I encountered the same bug. Logout and login won't fix it. Abandon the quest. Start the quest again. Kill the objective again. As soon as he dies, DO NOT waste time looting. RUN right behind the mouse up to the spot where he stands outside. This triggers the questgiver NPC to talk. The mouse then runs away, and you get credit.
Quest
Mouse Rescuers
A World of Warcraft quest in Zaralek Cavern. Facts and objectives from the current retail game data.
The highest voted Wowhead comments for this page. They are community notes, not TheWoWDB comments.
Questline is not tracked on wowhead as a storyline, so if it helps, these are the required quests. Final quest offers a 1000 Niffen rep token.
Mouse Detective
Sporemoth Exterminator
Agrulculture
A Friendly Introduction
[quest=74861]
Mouse Rescuers
Saddened that I appear to be the first person apparently OLD enough to recognize and comment that this quest is almost certainly a nod to the Disney movie 'The Rescuers'.
Quest facts
The client quest POI data records no map location for this quest, so no map is shown.
Check whether you have already completed this quest
in-game: /run print(C_QuestLog.IsQuestFlaggedCompleted(75462))
Description
I feel bad for everything I did working for Malrogg. He was just so big and scary, I never felt like I had a choice... <Agrul slowly shakes his head.> No more excuses. We need to get that sniffin' mouse back home to Yarren. I was never strong enough to take on Malrogg alone. But you... You can save Persii.
Community notes from Wowhead · 16 comments
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Reward details
Items you receive
Currency: Flightstones x 15
Earned from completing world quests, raids, dungeons, battlegrounds, arena matches, and other activities. Used to upgrade max-level gear.
Item facts are the current retail client’s (ItemSparse); the specialisations for a pick-one item are the ones the Blizzard quest API lists; faction and currency text is the client’s own. Lowest listed price = the US commodity auction house now.