I'll nominate Lush, lead singer Miki Berenyi was allegedly an avid Game Boy player and there's even a couple of pics of her and her Gameboy, one of them taken in backstage at Lollapalooza '92:
https://twitter.com/berenyi_miki/status/1010232773276262400
This is a Nintendo forum, need to say more?
Shoegaze bands tend to be overlooked for the most part, and Lush is one of the most well known shoegaze bands in the 90s.
Also since a lot of you guys seem have a thing for metal, there is a song by Lush that in my opinion is as metal as classic shoegaze can get:
Thanks for your consideration!