Hmmmmm......
Updates:
10/24/2013
Added 11 ideas for monster abilities - located in the first addition
If your monster has a positive status effect, it has immunity to 1 of it's weaknesses.
When your monster's hp falls below 25%, it's sp atk and atk is increased by 200%.
This monster has a stat boost for each monster alive in it's team.
This monster will never miss. (Moves have 100% accuracy)
Whenever an enemy monster deals damage to this monster, that monster gets a random negative status effect.
When this monster dies, it's opposing monster loses 15% of it's hp.
Maybe there can be abilities which support 2vs2 monster battles. "This monster's adjacent monster has a 10% hp stat."
edit:
Whenever this monster kills another monster, it is healed for 15% of it's hp.
Whenever a monster in this monster's team dies, this monster gets a 10% speed boost.
This monster deals 15% more damage against monsters of a lower rarity.
Whenever this monster uses a poison type move against an enemy monster, that monster becomes poisoned.
On your turn, this monster is healed for 5% of it's hp.
Enemy monsters have -1 in a random stat.
When this monster dies, another monster in your team gets whatever stat boosts this monster had.
Whenever this monster kills another monster, it gets a 25% exp boost at the end of the battle.
if this monster is your lead monster, you get a gold boost after each battle.
1st addition - added on 10/24/2013
(1) battle charge
Upon switch-in/switch-out, this monster deals 50 damage to it's opposing monster.
(2) heroic downfall
Upon death, the opposing monster loses 100 hp.
(3) oceanic blessing
Whenever a water type attack is used on your monster, your monster is healed equal to the damage that would have been done.
(4) assassin rush
All of this monster's attacking moves have +1 priority.
(5) treasured
Whenever an ally monster is killed, this monster gets a +5 stat boost in a random stat.
(6) flyaway
This monster takes less damage from flying type attacks and is immune to ground attacks.
(7) demonic demise
Whenever this monster kills an enemy monster, that monster's status effects is transferred to the next monster that the opposing player sends in.
(8) endure
If this monster is to die, it survives at 1 hp. This ability can only work once per battle.
(9) artful study
Your monster has 10 more MP for each move.
(10) dark promise
Upon death, the opposing monster is cursed to die in 4 turns.
(11) body building
This monster's max hp is increased by 25%.
10/26/2013
(12) flashback
This monster has a 25% chance of taking an extra turn.