Mikey showed up in "The Battling Eevee Brothers" as a one-off character. He's basically known for being caught in the middle of his brothers' constant arguing about how his Eevee should evolve.
All three of his older brothers had different ideas—each wanting him to train their favorite Eeveelution instead. Thing is, Mikey actually didn't want to evolve Eevee at all. He liked it exactly as it was. But he was too nervous to stand up to his brothers and tell them that.
Misty talked some sense into him, encouraging him to just be honest about what he wanted. When Mikey finally spoke up, his brothers weren't thrilled. It wasn't until they saw his Eevee hold its own against Team Rocket in a real battle that they finally got it and backed off. Seeing the Pokémon win without evolving proved Mikey's point better than any argument could.
Content compiled by AnimeList.moe from publicly available sources.
