This is one of the trickier quests in the game, so here's what you will need to do:
Talk to Martha and get the cat quest (you can also confirm on your map that the quest is being tracked)
- Pick up the fur, and bring it to Martha, which will make her very mad!
Be sure you have collected some feathers and fresh meat from animals on the island, and combine the fur with a feather in your inventory
Speak with Martha once again, present her with the newly stuffed cat, and receive the battery, completing the quest!