In the Nintendo DS version of Final Fantasy IV, you can find four optional eidolons for Rydia to summon through item drops.

GOBLIN can be obtained by the following monsters:

0.4% drop from Goblin (#002)
0.4% drop from Li'l Murderer (#148)
0.4% drop from Mopingway (#198)

COCKATRICE can be obtained by the following monsters:

0.4% drop from Helldiver (#004)
0.4% drop from Cockatrice (#026)
0.4% drop from Fledgling Rukh (#070)

BOMB can be obtained by the following monsters:

0.4% drop from Balloon (#105)
0.4% drop from Dark Grenade (#129)

MINDFLAYER can be obtained by the following monster:

0.4% drop from Mindflayer (#056)