So, I'm posting this question here because I am a beginner, and I suspect this has a very simple answer that my beginner-y mind won't jump to.

1.e4 e6 2.d4 d5 3.Nc3 Nf6 4.Bg5

What is the problem w/ 4...h6? And what would a sound response to that be?

As I said, probably a dumb question, but any thoughts would be much appreciated!

I'm also a beginner, but (or maybe because of that) I think there is no problem with h6 Fritz database reveals that (for example) Alekhine played h6 against Capablanca

Process of elimination. Don't try to figure out what the "book" move is. Simply look at moves and see what your options are.

A) 5.Bh4 g5 6.Bg3 just loses the pawn on e4, although ... de 7.f3 might be interesting.
... 6.e5 gxh4 7.exf6 Qxf6 similarly just loses a pawn

B) Other bishop retreats just drop the pawn on e4.

so that leaves

C) 5.Bxf6 Qxf6 when 6.e5 looks reasonable for white. You can support your e5 pawn with an eventual f4 because you're no longer entombing your dark-square bishop. This is probably what I would play.

But the thing is, if you're struggling with what to play here, you need to work on your fundamentals because you're going to be stuck constantly. What's your process for coming up with moves?

I have played a similar line: 1. e4 e6 2. d4 d5 3. Nc3 dxe4 (3. ... Nf6 4. Bg5 dxe4 5. f3 is similar) 4. f3. Black could take on f3 and then transpose into the Euwe variation of the Blackmar-Diemer Gambit, but 4. ... c5 is an idea.

7. ... c5 also seems like a reasonable idea against that line.