"Prince Caspian: The Return to Narnia." The reason is expressed in the often-forgotten subtitle.

I think the emotions the Pevensies feel when they return and discover that Narnia is not what they expected are shared more deeply by the reader if it is also our return to Narnia. We are in there shoes more.

I also think the 1300 year time jump is more shocking (as it should be) if you read PC as book #2. If you read it as #3 or #4, you've already done a little time jumping.

It all depends on whether you're reading chronologically or publication order. MN should always be followed by the LWW if you're following the timeline... but if you're reading in publication order (for the first time, mind) I think PC is the best second choice (as opposed to second best ), mainly for the reason Glumpuddle just gave, but also because we keep the progression of Narnia's creation intact, with it starting out kind of simplistic at first and gradually getting more solid and complex. If you read HHB second, and then go to PC, you might experience a slight change in the overall tone and style. You go from the long (but fascinating) sequence of political maneuvers in HHB to a more fairy-tale-ish, simple story in PC, which might slightly confuse new readers.
I say all this, but again, I just read whichever book I feel like at the time.

Well, Prince Caspian is the most logical choice but when I was a kid, I tried reading it after The Lion, the Witch and the Wardrobe and I got bored and didn't finish it.

After that, I read either The Magician's Nephew or The Horse and his Boy and I loved them. Sometimes I wonder if the reason most people aren't familiar with the Narnia books besides LWW is because they try to read PC, get bored and don't read further.

I guess if I were you, I'd start reading Prince Caspian to them and ask them if they wanted to switch to another book if they get bored.

P.S.
I really do like Prince Caspian. Honestly.

when I was a kid, I tried reading it after The Lion, the Witch and the Wardrobe and I got bored and didn't finish it.

Yeah, I've heard a lot of people say the same thing. I kinda struggled with it too towards the end. And that might be a good enough reason to save PC for later, I'll admit.

I think I would choose PC for a first-time read-through, if only for the reasons Glumpuddle mentioned: the readers' emotions track better with the Pevensies if this is the second adventure. I think having HHB come in later (after SC, where the characters first receive a mention) and MN right before LB both work really well, even if they do break up Eustace's stories a bit.

So my vote is stick with Publication order.

I think another benefit to reading in Publication Order is that HHB is a neat surprise to the reader. The first four books progress chronologically and you don't expect to learn much more about the Pevensies' rule in the Golden Age, so discovering HHB is kind of like finding a hidden tome from the distant past. Lewis teases so many little stories in CoN that we can only wonder about today, but the tease about HHB in SC actually pays off!

Also, for those who listen to Talking Beasts, think about Rilian's — the podcasting prince Rilian — experience of wonder when he discovered there was a book about the beginning of Narnia. If you read HHB straight after LWW, it's not going to feel like an amazing archeological find. And the same goes for MN, too. Reading those two later in the series makes you appreciate them a lot more, I think.

I think having LWW read first is the main reason why I'm in favour of introducing new readers to Narnia with publication order. Beyond that, I don't think it matters too much in which order you read the rest, provided that LB is last.

I think there's a logic to going with LWW > PC > VDT > SC because its chronological, but I could easily see MN working well if read earlier so that LWW is still fresh in the reader's mind.

I read MN after LWW, and it felt very logical for me to read the prequel. If memory serves me correctly, I read LWW AGAIN after reading MN, and then continued in publication order.