You can repeat the mission once per day. You may also delete the quest and wait approximately 10 minutes to be offered a new one.
The quest will ask you to capture 11 monsters from a random choice from one of the following categories. In brackets are some suggestions for monsters that should update the quest, but these are not meant to be an exhaustive list.
Capture several amphibians in an empty familiar cage. (Turtles and frogs, such as far east island turtles in lower Moors, muck toads in The Peat Bog or sand tortoises in Outpost of the Overlord.)
Capture several aquatic creatures in an empty familiar cage. (Crocodiles or crabs such as crabs in The Thundering Steppes.)
Capture several bats in an empty familiar cage. (Some may be found in Darklight Wood.)
Capture several birds in an empty familiar cage. (Hawks, vulrich, falcons and ducks. Some birds may be found in Antonica, near the oracle tower.)
Capture several boars in an empty familiar cage. (Pigs and boars will, but boarfiends will not update the quest. There are very few spawn locations for boars, so you may want to consider deleting the quest and requesting a new one in ten minutes. Alternatively, you can go to Nektropos Castle for swines (not the Swine Lord), and zone out and back in until you have 11.)
Capture several bovids in an empty familiar cage. (Cows, deers, sheep and camels. Deers may be found in Commonlands or Enchanted Lands.)
Capture several canines in an empty familiar cage. (Wolves and hyenas, such as timber wolves in Antonica.)
Capture several cats in an empty familiar cage. (Lions, tigers, panthers, leopards and sabertooths, such as the Stelecats in POM or lions in Commonlands.)
Capture several dire creatures in an empty familiar cage. (Raptors, dire bears, dire wolves, sasquatch or yeti, such as black wolves in Commonlands.)
Capture several drakes in an empty familiar cage. (Genuine, non Drakota drakes, such as the ones found in Greater Faydark. The quest is not doable on the Fallen Gate server before Echoes of Faydwer.)
Capture several enchanted creatures in an empty familiar cage. (Hellhounds, flying snakes and koalrat, such as blightrats in Rivervale, pond gliders at Crater Pond in Antonica or lashers in Enchanted Lands.)
Capture several mystical creatures in an empty familiar cage. (Manticores, cockatrice, displacer beasts, rust beasts, owlbears, such as rust monsters on the Butcherblock Mountains beach or duststingers in Commonlands.)