You might want to look into
Code Page 437, which was basically the de facto standard for PCs once IBM rolled around.
I remember reading about how the original designer of CP437 threw in card suits because someone mentioned that the computer should be able to play simple card games, and things like that.
The smiley faces were useful for roguelikes and derivatives, like ZZT, because it could represent the player. Various other symbols served other purposes, like the male symbol representing bombs, female symbols representing keys, and other things like that. Most of the greek symbols at the bottom row served as enemies, and they're pretty distinct looking too.
I think this'd be a good place to start. You probably don't need as many box-drawing characters though, so you could replace them with other things, such as chess pieces. Same with the greek symbols and the multitudes of accented letters.
Keep in mind, people had very good imaginations back then, if an entire character set of accented letters and greek symbols could be used for gaming objects. However, if your character set is
focused on gaming (instead of it just being a bonus), then I'd take into account several popular genres.
Namely, I'd think of Space Invaders, Gradius (and any other one-way-scrolling shmup), Lode Runner / Donkey Kong, and Roguelikes. So, that'd probably be a ship (optionally additional ship characters for facing in more than one direction), some kind of humanoid (so maybe a smiley, or a figure), and a set of "invaders" that could continually be reused in other games to provide a varied amount of enemies (or just "actors" in general). You'd need some collectibles and effects, so coins, jewels, rings, keys, bombs, explosions, bullets, things like that.
In addition, you'd need your standard ASCII set, box drawing, arrows, and maybe tiles dedicated to semigraphics (like in PETSCII), and I don't think it'd be a bad idea to include parlor game pieces, so card suits, chess pieces (which could also double as actors in other games), things like that.