First off, remember when Aunt Marge tells Harry, “you’d have gone straight to an orphanage if you’d been dumped on my doorstep”? And in reply, Harry thinks he would much rather live in an orphanage.  This is perhaps meant to subtly remind us that Tom Riddle grew up in an orphanage, which we learned at the end of Chamber of Secrets — another nice parallel between Harry and Voldemort.
Second (so far — I’ll be updating this post as needed), remember when Harry gives Stan Shunpike an alias on the Knight Bus? Is it mere coincidence that the first name he thinks to say is Neville Longbottom?  As I noted in my previous post on the subject, we will later learn that Harry and Neville have a lot more in common than was apparent at the beginning.
[Update:] And here's yet another subtle Harry / Neville connection. Remember the Quidditch finals? Harry can't sleep well the night before, tossing and turning with a series of nightmares about the final match. In one, he oversleeps the match, and when he finally turns up, Oliver Wood roars that they had to use Neville instead. Nice! 
 Rowling, J.K. Harry Potter and the Prisoner of Azkaban. New York: Scholastic, 1999, pp. 23-4.
 Ibid., p. 34.
 Ibid., p. 302.