Mmm, this is tricky. It's a matter of whether a change of tense is required in reported speech. I judge that it is (for the cooking/cooks part). I would opt for:
Bobby visited his grandma yesterday. Grandma asked if he would enjoy playing in the game room while she cooked.
I suspect the answer is both. You see, it probably depends on what she said. If she said 'while I cook', use 'while she cooked'. If she said 'while I am cooking', use 'while she was cooking'.
In my opinion, she could have said either and been correct. So, both answers are right methinks.