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 one of the following categories, in an empty familiar cage that you receive from the Steward. Below are some suggestions for monsters that should update the quest, but these are not meant to be an exhaustive list.
Birds: hawks, vulrich, falcons and ducks. Examples include birds found in Antonica, near the oracle tower.
Boars: pigs and hellboars. Note that 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.
Pigs also spawn in Thundering Steppes (Thundermist Village), and in Commonlands (Crossroads). These pigs have bags above their heads, but unfortunately you cannot capture them. Hopefully this will be corrected!
Canines: wolves and hyenas. Examples include timber wolves in Antonica.
Cats: lions, tigers, panthers, leopards and sabertooths. Examples include the Stelecats in POM or lions in Commonlands.
Dire creatures: raptors, dire bears, dire wolves, sasquatch or yeti. Examples include black wolves in Commonlands.
Drakes: drakes or fae drakes. Note that drakota will not update the quest, even if their name or appearence may indicate they would be drakes. The quest cannot be completed on the Fallen Gate server until the Echoes of Faydwer expansion is enabled.
Enchanted creatures: hellhounds, flying snakes and koalrat. Examples include blightrats in Rivervale, pond gliders at Crater Pond in Antonica or lashers in Enchanted Lands.)
Mystical creatures: manticores, cockatrice, displacer beasts, rust monsters and owlbears. Examples include rust monsters on the Butcherblock Mountains beach or duststingers in Commonlands.)
Reptiles: chokidai and lizards. Examples include lizards or monitors in The Peat Bog, Commonlands ( 114, -43, 55 ) /waypoint 113.62, -42.70, 54.95 and ( -38, -46, 243 ) /waypoint -37.86, -46.47, 242.87 or Antonica.)