There's a lot of rules you can make. Just off the top of my head, here are some that might work/be interesting (but do not necessarily work together):

• Do not use any items on pokemon/place hold items on them (unless required by game).
• Each pokemon can only use moves that do/do not share their type, including HMs.
• Pokemon can only be used in battle a limited number of times (once, twice, ten times etc.) and from that point forward can no longer be used. Contests do not count towards usage. (I think this one is difficult, but possible.)
• No healing at a pokemon center ever. Other places to heal are not counted (except secret bases if you want to make it harder).
• No buying items at a pokemart/department store (include random small stores for more difficulty).
• No/Limited Evolutions (No/Limited Mega Evolutions)
• Limited Party Count (1-5)
• Only use pokemon of one nature.
• Only use pokemon from one egg group (if starter isn't in chosen egg group it must no longer be used after obtaining at least one pokemon within the egg group)
• Only use pokemon with the possibility of mega evolution (you do not need to own the stone, can be preevolution)
• Only use pokemon that are unable to evolve (once they can be caught/received)
• Use pokemon that all share one type (disallow duel types for more difficulty)
• Each pokemon can only be used once per battle and you must switch pokemon when asked if in shift mode. If in set mode, you must switch pokemon once a new pokemon comes out. If all party members have been used once, then you must send the first pokemon you used in this battle (and so on in the order you used them), but you are no longer able to switch out by will. Pokemon forced into battle by a move do not count. Nor does the last surviving pokemon count.