Does 给你 mean "Here you go?" I found a very short sentence on tatoeba.org. All it says is, "给你。"

That literally means "Give you," but what does it really mean? My best guess is that I could say it as I give something to you, kind of like, "Here you go."
Your guess is correct. You use it when you hand over or give something. You can translate it as "there you are, here you are, here you go".

For example a teacher might say it while handing out some study material. A student of course should use 给您 when handing in his exam papers.

You can also add what you are giving. For example 给你一本书.