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!