Simply, if you brother really loves games then to be honest he shouldn't be giving them up for a girl and if his girl really loves him she shouldn't be asking him to give up something that he loves.
I don't think a relationship that necessitates you to change part of who you are or what you love doing is a healthy one. My advice is that he should discuss this with her and explain his position and perhaps they can at least come to some sort of compromise, for example less games or no game playing when she's around or something.
If she won't budge and insists that he gives up games it begs the questions, what is she going to ask him to change or give up next and what compromises is she making for the sake of the relationship?
Failing a jolly happy compromise, what Twesterm said.
