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.
Boars: pigs (bestiary) and hellboars (bestiary). 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!
Drakes: drakes (bestiary) and fae drakes (bestiary). 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.