I would like to send someone who already has a Steam account a money gift so that he can buy himself some games.
Is this possible? If so, how?
There is no option to send money to the wallet of your friend directly.
What you can do however, is buy a Steam gift card at GameStop and send that code to your buddy so he can activate it. I do not know if that works cross countries though. But if you both live in the same country and GameStop has those gift cards available, then it would be no problem to activate it.
Another option would be to put an TF2 item on the Steam Market and you buy that. Valve takes a cut here though, so it is not really viable and I don't recommend it.
The best option imho would be just to ask him what game he wants and gift that directly.
You can send your friend money via PayPal. Steam accepts PayPal as a form of payment.
so that he can choose himself some games.
Slightly redefining the stated problem, that's what the Steam wishlist is for. If you have him as a friend on steam, and he has items on his wishlist, you can gift him those.
Steam recently added the ability to buy digital gift cards online, which is effectively what you need here.