I don't think there's much of Lewis' work that wasn't at least partially based on his life– had it been Tolkien, this would be open and shut case, but Lewis was much more comfortable with intertwining real life, allegory, parallels and supposals into the nooks and crannies of Narnia. But unfortunately, we're left to guess at most of those supposals that Lewis didn't really talk about. The famous allegorical quote in Lewis' letters speaks to this:

If Aslan represented the immaterial Deity in the same way in which the Giant Despair represents despair, he would be an allegorical figure. In reality however he is an invention giving and imaginary answer to the question "What might Christ become like if there really were a world like Narnia and He chose to be incarnate and die and rise again in that world as He actually has done in ours?" This is not an allegory at all

- The letters of C.S. Lewis, paragraph 6, page 283.

I'd be comfortable with the first three, but the fourth as you said is a stretch, considering that HHB was probably written in 1952-54, and published in '54, and Lewis and Gresham were married in '56. I know they wrote previously, but I'm not sure how serious that relationship was until much later.

Posted by: @cleander

1. Shasta's longing for a distant, unknown world, much like Lewis' early longings. 

I think there are some specific links to Lewis's childhood here, mainly in the form of the green hill in the opening chapter:

But he was very interested in everything that lay to the north because no-one ever went that way and he was never allowed to go there himself. When he was sitting out of doors mending the nets, and all alone, he would often look eagerly to the north. One could see nothing but a grassy slope running up to a level ridge and beyond that the sky with perhaps a few birds in it.

Sometimes if Arsheesh was there Shasta would say, "O my Father, what is there beyond that hill?"

And from the first chapter (coincidence???? ) of Lewis's autobiography:

And every day there were what we called "the Green Hills"; that is, the low line of the Castlereagh Hills which we saw from the nursery windows. They were not very far off but they were, to children, quite unattainable. They taught me longing—Sehnsucht; made me for good or ill, and before I was six years old, a votary of the Blue Flower.

That's always stuck out to me as a pretty direct connection with Lewis's childhood memories.

Posted by: @cleander

4.(Stretching this one a bit) Possibly Aravis was inspired by Joy Gresham???

@ericnovak is right that the timing doesn't really check out, but that's an interesting thought. From what I know about Joy's personality, Aravis seems to be quite a bit like her, so one does wonder if there was any influence there at all. According to the timeline on Narnia.com, Lewis began and completed The Horse and His Boy in 1950, and that's when his letter correspondence with Joy began; they didn't meet until 1952, and the book wasn't published until 1954. So yes, it's a bit of a stretch, but interesting to think about, still.

Actually, I'm reminded of something that Douglas Gresham has said in a couple of talks that I've listened to: that Lewis had a peculiar knack for writing characters in books he was going to meet in real life. Douglas has always said his personality was a combination of the Archenland twins, but I wonder if he might also be referring to Aravis and his mother Joy?

I had not considered the timing issue concerning Jack meeting Joy. (Apparently neither had Kathryn Lindskoog.) That's interesting though, Rose, about Douglas Gresham saying Lewis had a knack for writing about people he was going to meet. It does open up a vast new world of speculation!!! Who else might that apply to?  

Thanks for the text comparison as well! I'd forgotten Lewis mentioned green hills in his autobiography- that fact makes the parallel even more noticeable. 

Much like the "Planet Narnia" theory, i think this sort of analysis only works in the confirmatory sense - i.e. there are probably certain elements you can find which fit the theory, as long as you disregard all evidence which doesn't fit the theory.

For example, if you look at the major milestone events of C.S. Lewis' life, such as fighting in the trenches of World War 1, being a prominent scholar at one of the world's most prestigious universities, becoming a world famous published author, etc. then I don't think you could say that comparable events happen to Shasta, either in a literal sense or metaphorical sense.

Equally if you look at the central components of Shasta's story, such as being raised by a man who wasn't his father, or having a long-lost identical twin brother, then again I think you would struggle to find parallels from CS Lewis life, metaphorical or otherwise.

I know you qualified it in the original post that you were only talking about it being semi-autobiographical, but I think once you get into the realms of only applying a vague thematic parallel on selected elements, the entire notion breaks down, and as you say that is ultimately no different to every author who uses certain experiences from their own lives to express their own emotional truths within the story.

@icarus, Your point is well made. While I think it's fair to say that this theory does tend to only work in the confirmatory sense, it is, unlike Planet Narnia, emphatically not a serious theory but rather more of a speculative one. 

I still hold that there is a little more clear evidence of Lewis possibly personally projecting himself in this story compared to all the others, but I doubt you could build a compelling case that "Shasta is really C. S. Lewis, etc."

(On a slight tangent, I believe one instance in C. S. Lewis' life which you mentioned does find a slight parallel in the book- that of the experience of war. It's still debatable however if Shasta's experience of a medieval cavalry battle has anything to do with Jack's experience as a soldier in WWI.)

Overall, I do not think that Lewis sat down and decided to write his autobiography disguised as a children's story. The book is autobiographical only in the sense that it obviously and closely reflects some moments in the author's life, to an extent which is not present in the other Narnia books. I doubt that this is by design, nor would I assert that this book contains some "hidden meaning" that we're not supposed to know about. 

Another  good example of what I'm suggesting would be Charles Dickens and his story of David Copperfield. Many events in the character's life are inspired directly by Dickens's experiences- such as working in a blacking factory at a very young age.The story does not exacrly parallel Dickens's life- it wasn't intended to-  but it contains parallels which the author openly pointed out. 

I wouldn't say The Horse and his Boy isn't autobiographical. I'm just not sure it can be described as more autobiographical than the other Narnia books. A lot of them cover the themes Cleander mentioned in the OP. That being said....

Posted by: @icarus

For example, if you look at the major milestone events of C.S. Lewis' life, such as fighting in the trenches of World War 1, being a prominent scholar at one of the world's most prestigious universities, becoming a world famous published author, etc. then I don't think you could say that comparable events happen to Shasta, either in a literal sense or metaphorical sense.

This is assuming that C. S. Lewis himself would have considered those the major milestones in his life. It's natural for us to assume they were of course. Being a prominent scholar at one of the world's most prestigious universities is palpably impressive and World War 1 infamously was so miserable that it left many people emotionally shattered and disillusioned. But just because it's natural for us to assume something about how historical people perceived their own experiences doesn't mean it's true. If I remember right, in Pilgrim's Regress, which explicitly was autobiographical, Lewis doesn't focus on those things that much.