Weight does seem to matter. If you feed them too much, they turn into the Large / Sumo family. If you feed them too little, they turn into the Puchi / Petite family. Having said that, I don't think it's the weight that's tracked, but rather how many hungry hearts are full, on average. On my first generation, I fed only a few snacks, and played lots of games, but I still got the Large / Sumo family. I think this was because I kept the hungry hearts all full, as much as I could. In generations since then, I've always only filled up to three hungry hearts, and haven't had any problems.