I believe that the Beavers explained it, "'Why, all she wants is to get all four of you...But she'll keep him alive as long as he's the only one she's got, because she'll want to use him as a decoy; a bait to catch the rest of you with.'"
She probably wanted to get rid of them all to make sure that they did not have any hope of taking the throne, although it seemed as if she would want to turn them into statues. At least, that's what Mrs. Beaver reasoned. If I was her, I would want to kill them all, but then again it is unclear as to whether or not she knew that Aslan could bring people back to life after they were turned into stone. She really could have stopped the prophecy if she had just killed Edmund at first.
However, after Edmund told her that Aslan was in Narnia again, she probably saw it as an opportunity to kill Aslan, and then later kill Edmund.

Hast hit it on the head, friend Lilygloves. But I'd like to suggest some alternative ideas.

1. Jadis is lonely, having found out that 'a high and lonely destiny' is not what it is cracked up to be.

2. Having killed off her whole world, and found herself in a hall of memorial statues and nobody to want to remember her, she only kills if she can insist it is their own fault that they got killed.

Jadis is lonely, having found out that 'a high and lonely destiny' is not what it is cracked up to be.

This does not make much sense to me logically. Where, anywhere throughout the books, does Jadis show some sort of ability to feel such a sensitive emotion?

She really could have stopped the prophecy if she had just killed Edmund at first

Which is what doesn't make sense. A Pevensie in the hand is worth four in the wardrobe. . . right?

Jadis is lonely, having found out that 'a high and lonely destiny' is not what it is cracked up to be.

This does not make much sense to me logically. Where, anywhere throughout the books, does Jadis show some sort of ability to feel such a sensitive emotion?

In MN she does. In both MN and in LWW she plays the role of temptress which shows she has the ability to understand human emotional patterns, such as appealing to Digory's anxiety for his mother, or to Edmund's resentment of his siblings, even if she doesn't feel such a sensitive emotion as loneliness, herself.

I doubt she is lonely in the sense you and I could be. From the time of her realisation that she had spilled the blood of all her own people, and after she had destroyed her sister's army, and therefore all Charn, Jadis wasn't awake and thinking whilst she was in a charmed sleep in Charn.

The way she expected to be awakened is spelled out on page 63 of MN (my copy), on the last page or the second last page of Chapter 5. Jadis is so full of herself, she expects that Uncle Andrew is some great king who has seen her beauty in a mirror and has called to her across the ages and the gulf which divides them. In other words that she would appeal to somebody or other - her soul mate you might say. It seems even Wicked White Witches need soul mates, perhaps?

My other reasons show why Jadis would logically be lonely without people around, or the sorts of people she can identify with, eg humans or royalty.

1.Nobody to show off to, just how magnificent she was. Nobody to fear her, talk to her or interact with. Also she can't live in the style she had been expecting if nobody around to be her minion. Why she wants to escape to Polly and Digory's world, and why she didn't simply turn everyone into stone when she first returned to Narnia to cause the 100 years of winter.

2.Her weakened reaction in the Wood of the Worlds where she would be left alone in peace and quiet, if Polly and Digory had succeeded in leaving her there.

3. Jadis' desire to be immortal by eating the apple in the garden succeeds, but she is thereafter deathly white having found 'her heart's desire and found despair', to paraphrase p. 146 of my copy of MN.

4. Aslan's remarks at the end of that book that 'length of days with an evil heart is only length of misery and already she begins to know it'.

Well, I think the reason she didn't kill him at first was simply because she wanted more information. If he knew how to get in, did others? If so, how many others were there? Once she had found out that there were four of them and (according to Edmund) they were the only ones who knew about Narnia, she wanted to kill all four them. Getting all four of them was particularly important not only because of the prophecy, but also because Edmund tells her Lucy had already met a Faun. If she didn't get all of them the Narnians might protect the others and she would always have a possible uprising on her hands as long as at least one human was still alive in Narnia.

Then, when Edmund comes back alone, she has no need to kill him right away because he is a traitor. She thinks she can catch all of them when Edmund tells her his siblings are at the Beaver's, but then he mentions Aslan. She wants to make sure the others do not reach Aslan and thus has no time to bother with killing him. It is only after she knows she cannot get the others that she is ready to kill Edmund. She's also a bit of a stickler for tradition, which is probably why she delayed a little. She wanted to kill him on the Stone Table. "That is the proper place. That is where it has always been done before."

Don't the villians always crave more (more power, more wealth, more destruction...)? Why would Jadis simply take out Edmund when there were three more pieces to the prophecy? Wouldn't killing all four humans be more satisfying for the White Witch than just one? For her, it seems to me, killing Edmund right off the bat would be too easy. She's an evil witch with a manipulative brain. It is far more clever to draw your enemy in (Edmund for example) and use them to get what you want (or even more of an interesting alternative- turn him against his own family). Let's not forget her manipulative words used on Digory back in MN - why not turn innocence into something destructive (Digory ruining his mother's life by steraling the apple and having to live with the agony all his life or Edmund's act of a traitor).

Perhaps, for Jadis, there is more satisfaction in seeing your foes destroy themselves rather than killling them from the start... another example in her wait to use the deplorable word after Charn was left in waste and ruin from war.

All of these answers above are good.

The main reasons I think Jadis did not kill Edmund on sight was that:
1. She wants Ed to turn evil.
2. She is vain and wants to have statues of four children to show off to her evil cohorts that she will always be queen.
3. If she killed Edmund in the woods, someone might see her and have a convicted conscience. It would be one thing for the local inhabitants to hear about her murders, but would be another thing for them to actually see them. Notice that Hitler shipped the people he did not like away before killing them.

2. She is vain and wants to have statues of four children to show off to her evil cohorts that she will always be queen

Well, yes.....I agree that Jadis is vain. But could she really turn human children into statues? In MN, where we see what Jadis could do, in different worlds, we also see what she could not do. And, unlike the other inhabitants of Narnia, as Jadis well knew, humans weren't native to Narnia, any more than she was, herself. They are native to this world, as Jadis was native to Charn. Here, in this world, Jadis was immensely strong, but she could not use the 'deplorable word' on Aunt Letty, who merely thought that Jadis was using strong language (swear words). She couldn't use such language on the mob or policemen, either, could she?

Well, I think the reason she didn't kill him at first was simply because she wanted more information. If he knew how to get in, did others? If so, how many others were there?

Yes, this is the most logical reason why Jadis would not have wanted to kill Edmund straight off, or Lucy, for that matter. By encasing Narnia in a 100 year's winter she had stopped anyone getting into Narnia from the Wood between the Worlds, the way she came into Narnia, herself. And she knew that Uncle Andrew, Polly Plummer and Digory Kirke had left Narnia, probably to go back to London. She couldn't know how much time had passed in the other world, where the rings were, or who else those three humans might have told about Narnia.

Once she had found out that there were four of them and (according to Edmund) they were the only ones who knew about Narnia, she wanted to kill all four them. Getting all four of them was particularly important not only because of the prophecy, but also because Edmund tells her Lucy had already met a Faun.....She's also a bit of a stickler for tradition, which is probably why she delayed a little. She wanted to kill him on the Stone Table. "That is the proper place. That is where it has always been done before."

Yes, thwarting the prophecy about Adam's flesh and Adam's bone in Cair Paravel enthroned is her main consideration. Apparently, like Voldemort in HP, she actually believes this prophecy. But what would have happened if the White Witch had met Lucy first? Would she have killed her outright then? Or was there something else constraining Jadis, which forces her into being a stickler for tradition? The other prophecy about Aslan, for example?

She wants to use him as bait, to lure his siblings, so that they don't reveal the door to anyone else. I'm guessing she feared that even if she killed Edmund, that his siblings would have brought someone else through the door to be the new 4th human. If she killed all the kids, no one else might ever come in from their world because they won't know the way.

I think Jadis wanted to use Edmund to lure the rest of the Pevensies to her. If she had killed or hurt Edmund then I think she would have given herself away and then she never would have gotten them (even though in the end she still didn't).
She wanted to use her deceit and trickery to get them to come to her.
That was my thought anyway.

Maybe it just didn't really cross Jadis' mind (after all, even witches are allowed to be a little absent minded from time to time). I remember in the book at one point she suddenly thinks "if I kill him, then the prophecy won't come true' (okay, it wasn't really worded like that, but I'm a little to lazy to pull out my Narnia book and write the exact quote) and then she almost kills Edmund but the Narnians come and rescue him just in time. Maybe before it didn't really dawn on her that if she's killed one, then that will stop the prophecy from coming true. Maybe it sounds sort of silly, but that was sort of the way that I looked at it. In the beginning she wanted them all killed, cause after all, even one of them could, in some way, pose a threat to her rule, and she thought it would be easy (seeing as Edmund wasn't really the brightest of kids at that point) to just get them all, especially if she appeared to be nice. But then things didn't turn out the way she had hoped and as they got worse and worse, she realized (or decided) that even only one of them dead would be good enough.
That's just the way that I see it

I beleive that the prophesy could not just be fulfilled by any four children, only by the Pevensies. But perhaps she wasn't sure that these were indeed the four children spoken about in the prophesy. Only if she was positive would it have made sense for her to kill Edmund right away (or at least after he had told her about his brother and two sisters), because it is stated in the book and movie that all four children were needed to reign in Cair Paravel.

I think the book is quite clear on this point:

[Mrs. Beaver:] “Why, all she wants is to get all four of you (she’s thinking all the time of those four thrones at Cair Paravel). Once you were all four inside her House her job would be done—and there’d be four new statues in her collection before you’d had time to speak. But she’ll keep [Edmund] alive as long as he’s the only one she’s got, because she’ll want to use him as a decoy; as bait to catch the rest of you with.”

--LWW, Ch. 8

She changes her strategy after realizing she won't be able to prevent the other three children from reaching the safety of Aslan's camp at the Stone Table...

“No,” said the dwarf, “it is no use now, O Queen. They must have reached the Stone Table by now.”
[...]
“Four thrones in Cair Paravel,” said the Witch. “How if only three were filled? That would not fulfill the prophecy.” 

[...]
“Yet it might be better,” said the dwarf, “to keep this one” (here he kicked Edmund) “for bargaining with.” 





“Yes! and have him rescued,” said the Witch scornfully.

“Then,” said the dwarf, “we had better do what we have to do at once.”

--LWW, Ch. 13

It was a desperate move, but at that point, what choice did she have? Aslan and the children clearly had the advantage, so she had to do something.

I beleive that the prophesy could not just be fulfilled by any four children, only by the Pevensies. But perhaps she wasn't sure that these were indeed the four children spoken about in the prophesy. Only if she was positive would it have made sense for her to kill Edmund right away (or at least after he had told her about his brother and two sisters), because it is stated in the book and movie that all four children were needed to reign in Cair Paravel.

For reasons of the story, and Aslan's plan, that might make sense, but logically, no.

Nothing in prophesy ever mentioned that they had to be blood related at all. Aslan could have chosen four random kids form differnt parts of the world to come to Narnia, but Pevensies made sense.

As long as the other three kids were still alive (assuming Ed was killed) Then they could go back and (in theory, althugh they most likely wouldn't) go back and get another boy to come along with them. That seems mostly likely to be Jadis's logic