This is one of the achievements that requires you to find all secret summons for Rydia. They are rare items dropped by monsters.
Once you get the item, open your "inventory -> select the item", then it'll automatically appear in the summon list.

First, go to Hummingway Home and buy 99 Siren.
Pro Tip: before farming these summons, save your game first, so when you run out of Siren, you can simply reload the game without going back to the moon to buy more Siren.

Goblin

Dropped By: Goblin
Location: Baron Area
Using Siren is guaranteed to encounter Goblin (x6).

Cockatrice

Dropped By: Helldiver, Cockatrice, Roc Baby
Location: Mysidia Area
Using Siren is guaranteed to encounter Zu, Helldiver (x3).

Mind Flayer

Dropped By: Mind Flayer
Location: Magnetic Cavern - B1
Using Siren is guaranteed to encounter Mind Flayer (x2).

Bomb

Dropped By: Balloon, Dark Grenade
Location: Moon - Western Lunar Path
This summon is probably the hardest since you cannot use Siren. Walk inside the Western Lunar Path tunnel and you'll fight one of the following encounters:

• Dark Grenade (x3), Abyss Worm
• Balloon (x2), Dark Grenade (x2)
• Eukaryote (x2), Prokaryote
• Eukaryote, Prokaryote, Abyss Worm
• Black Flan, White Mousse, Purple Bavarois
• Abyss Worm (x2)