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.
Note that boarfiends will not update the quest. Hellboars (bestiary) also do not appear to 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.
Examples include the linked swines in Nektropos Castle. You will, however, have to zone out and back in repeatedly to obtain enough spawns for the 11 kills. Note that the single triple-up swine cannot be caged, and the Swine Lord will not update the quest; the encounter must be the version with 2 or 3 linked swine.
To capture a monster, you need to be hated by the encounter, but you must not kill the encounter before you have used the cage.
Many heroic encounters, raid encounters and non-attackable NPCs (such as the pigs in Windstalker Village) may not be caged. Some of these monsters may nevertheless display the quest icon (sack) on their names.
If caging a low level monster, you may want to engage the encounter with your autoattack turned off (or not wielding weapons), with an ability that does not deal damage.