Is there any use for the hand that you find in the guest house?
Solution 1:
Regardless of picking up the hand or not, although most people do, when you are at the dinner table your hand is stapled back on. Even though it isn't realistic (but it is a game, with zombies, and first aid liquid etc), so yes Ethan shouldn't have any use of his hand after it is stapled back on but spoiler:
He's infected just like the Bakers are so that's the other reason why his arm is "healed" and again usable, that and it being a game and for the sake of progression it also works. The idea is that because he has the "Mold" he is able to coapt the limbs easily. The first aid liquid is supposedly something that just speeds up the process and helps you regenerate/heal quicker. -- You picking it up and then seeing it back on you working is supposed to hint to the fact that you are now infected.
So you can leave the hand if you want, it'll find its way back to you ;)
- To directly answer your question in short: The use for the hand is quite literally you regain usability. And the point of picking it up: There isn't much other reason than because you can.
You can see more in this discussion.