What sort of things do you think experts (professors, generally) might value that a less-expert person like myself might be able to offer?

Lots of things, of course: adoration, bacon, sexual favors, etc. etc. :-D

In practice, I suspect that some attention, gratefulness, and a demonstration that you're not a clueless idiot with some agenda will go a long way towards making the expert willing to answer your questions. The last part is the problematic one in online communications -- by default you're just "another guy from the internet" and we all know what the average of that looks like.

However something in this vein seems like not a bad start to me: "Dear Professor X, I read your papers/books Y and Z and was amazed how you figured out A, B, and C. However I have a question about D because while E it seems to me that F." Demonstrate cluefulness and use flattery :-)

P.S. Another important issue is scope. Ask questions that can be concisely answered in a couple of paragraphs. Do not ask questions the answers to which are a graduate degree, a shelf of books, and a really tall stack of printed-out papers (e.g. "What should I eat for health and fitness?").