I am of the opinion that all the Narnia stories are connected some how in some way. HHB is definitely connected with LWW in the Narnian Golden Age and Prince Caspian is the result of many years of Telmarine occupation. I personally feel that the books are the telling of an overarching story and as I understand more of the books and movies, the more I am convinced of it.

I voted Option 2, in part because I think that describing something as having an overarching story means it should be like Lord of the Rings or Harry Potter, in that the individual books are part of a single story, none can be removed without the story being incomplete, and events of one book affect the next.

To be sure, the Chronicles are not completely independent of each other, and I think they're best enjoyed together, but they can absolutely be enjoyed alone. I think some of the books, like PC and TLB, might suffer if you go into them completely unaware of the rest of the Narnian world, but you don't strictly speaking have to have read LWW to understand what's happening in PC. HHB I think works very well on its own, and not having read LWW or any of the other books could almost serve you, since the main character has also never been to Narnia before either.

I think another reason I prefer to think of them as 7 individual but related stories, instead of one large story, is that it makes the Narnian world feel more alive to me. The 7 books don't comprise the story of Narnia, they comprise 7 of the stories of Narnia. There's entire histories taking place between the books that only exists in hints and our imaginations.

I think of them as connected stories as well, simply because I can't imagine them all as one book. The style changes slightly throughout the series, as do the main characters. Also, HHB seems to be an oddball in the '' One Big Story" arrangement because LWW seems to skip the events related in that book.
For this reason, as far as future adaptations, I think long films are more the way to go than episodes. I'm really hoping the episodes will be reserved for expansions and non-canonical world-building stuff. (Having each book on one DVD, assuming they get released, would also be better [imo] than having them spread out over seasons.)
I vote 2!

I've always seen them as a series of related stories as each book had distinct plotlines and covered specific timeframes. Certainly, there is a continuity in lore, and each book either builds or contributes to another. But there isn't necessarily a consistent flow in time; or in other words, each book focuses on a specific time period, the series itself doesn't necessarily cover everything with its given timeline. I think we would have more books if Lewis had wanted to cover more of the world he'd created.

But I think I can see how some might see it as a big story told in parts. If I was looking at the country of Narnia alone as a history subject, then I probably would've chosen Option 1. And in The Last Battle, Lewis described the characters' lives in our world and their adventures in Narnia as only "the cover and the title page" of the Great Story which no one on earth has read. (The Chronicles of Narnia, Paperback - p. 767)

A series of books. I agree with what Cleander has said, that the style changes from book to book. I would add that that superficial story and character beats (defeating evil, boy/girl dynamics) are very similar between stories, mostly because they were written separately at stand-alone stories. Unlike with The Lord of the Rings, where that is the title of the story (and the Fellowship of the Ring, Two Towers & Return of the King names are all subtitles), The Chronicles of Narnia title is a brand, and the book titles are the real names. That’s why I’m so irritated at the Narnia movies being called ‘The Chronicles of Narnia: The Lion, The Witch & The Wardrobe’ instead of simply ‘The Lion, The Witch & The Wardrobe’ or even ‘The Lion, The Witch & The Wardrobe: The Chronicles of Narnia’.

The naming of the books 'The Chronicles of Narnia' defines them as a set of books, rather than one long ramble.
It is different from the Chronicles books in the Bible.

We can say that a story "chronicles the history of.." a person or place or group. This doesn't need to be one long story, but is usually a set of tales from within those years.

I think the seven Narnia books are seven separate stories since each one has a separate plot and they can be read independently. It would be hard for me to understand why someone would want to quit the series after reading only one or two of the books, but I guess some people may not have the same love that I do. It's hard for me to live without all seven. I would say say each book stands very well on its own so at least try one or two of them and you might be hooked as I was so many years ago.

I definitely see tham more as a series of related stories. While most of the books do feature a lot of the same characters and, for the most part flow naturally from one to the next. They are still fairly standalone(with the exception of LB and maybe PC.) And are connected more by theme then any overarching plot.

I have always seen Narnia as a series. Though it does have a similar plot sometimes in the books, I don't see it as one really continuous book. Narnia has always been to me a series because even though it always involves Narnia and sometimes the same characters, in every book there is a different problem to face and a different goal to achieve. Whereas if it was a continuing book simply split up into separate books each, then all the books would point to one direct goal, and not a separate adventure in each book. It would also involve the exact same characters throughout the whole series, but Narnia has different characters through each book. Each Narnia book is different in it's own way and even though HHB doesn't have the same characters or the same format or plot as the other books, it still feels like part of the series because it's still about Narnia. 

It's also worth noting the line in The Last Battle (I think?) in which Jill (or is it Lucy?) is complaining about how Narnia always is in war or tribulation, but then some Narnian or other (my memory on this section is shocking) tells her more, about Swanwhite and Moonwood and others, et cetera, thus implying that there is more, so much more, to Narnia; and this is where I think spinoffs could serve us well, if we got all seven books of course, to show that Narnia is more than one overarching plot, but is a whole country and nation!

I would categorize Narnia as a series of related stories.  As a kid I sort of expected it to be one overarching story, but that's just not how Narnia is presented.  Like okay, Narnia's original evil, the White Witch, is killed after the first/second book (depending on reading order).  After that comes a series of disparate villains - the Telmarines, the Calormenes, the LotGK and the false god of Calormen, the super-mysterious Tash, who makes a seemingly random appearance in LB.  Then there's Voyage, which doesn't really have a villain at all.  But enough about villains, let's talk heroes.  The Pevensies appear in most of the books, but some don't really do much with them.  They're mentioned in SC and VDT, and they're kind of involved in HHB (in a super-indirect way) but it's not all about them.  The one thing constant is Aslan.  And I think that's kind of the point. 🙂

I think the problem is our expectation of an over-arching story.  That's how series work in general.  Most of the fans I know, myself included, rank MN and LWW in their top 3, and I think that's part of the reason.  Like okay, if I were writing this series, I would have done a few things differently.  Like the other witch in SC would have been the same one as in MN/LWW.  It makes sense, gives her some motive for trying to conquer Narnia, and I could go on.  Calormen would probably have been the nation to conquer Narnia in PC (or Telmar would appear in other stories at least).  Tash would be more visible, and the final battle would be literally everybody vs. Narnia, led by the Witch.  The Pevensie kids would be more involved in HHB and SC, and I think there would be a lot more continuity in general.  I highly doubt it would be half as good as the Narnia we know and love, but that is what an over-arching story would look like to me.

But don't get me wrong - I like the more conventional over-arching story, with recurring main characters and logical continuity and all that - but Narnia being so different from that is a good thing IMO.  When every little detail is meticulously spelled out, there's nothing left to discuss.  There's no room for fun exploration of ideas, for theorizing, speculation, and stuff like that.  Not so with Narnia.  I love the unanswered questions, the plot-holes, and how at times it just seems... random.  I think it actually makes Narnia more real than other made-up worlds, because there's a lot we still don't know, maybe never will.

PS: My least favorite, to answer your other question, has always been the Last Battle.  I dislike endings in general (always sad to see a favorite thing be over) and that was intensified by having all the main characters DIE at the end.  I like HHB.  I wouldn't rank it as high as LWW, MN or SC, but it's somewhere in the middle for me.  It was a great story, and I'd rank it higher than VDT (but I love VDT too... lol they're all great, who am I kidding?)