The old man wants a low-level Pokemon to remind him of the energy of youth. You can freely take back the Pokemon at any time, and give him a different one, but if you ever beat the league while you've donated a Pokemon to him, the man will disappear permanently after that. If you return to his house, he obviously won't be there; instead there will be a Poke Ball allowing you to take your Pokemon back, and you get to read his letter thanking you for letting your Pokemon keep him company in his dying moments.
As for what the benefit is, when you take back your Pokemon this way, it gets its affection meter completely maxed (making this the fastest way to max out that stat in the generation, albeit with limited use) and you also get a Comet Shard to sell for P15,000, although the act of beating the league was already worth more than four times that much.
