Walkthrough

1. For completing this request, you can get Gimmighoul Trades as a reward.
2. Gimmighoul has the Collect specialty, allowing you to directly trade goods with them without monetary value like Trade. By exchanging either a Rainbow Feather, Silver Feather, or a Lost Relic, you'll get a random furniture item from Gimmighoul.
3. How to Trade Items With Gimmighoul
4. Go to the museum and head down to the magma area. Go north by either using Water Gun on the magma to make it walkable or use blocks to make a makeshift bridge to reach the area.
5. Enter the museum to meet Gimmighoul. Talk to them to befriend them.
6. Look for a Rainbow Feather, which you can find randomly as you explore. The best way to know that a Rainbow Feather is nearby is if you hear Ho-Oh's cry. Find the feather around the area and use Water Gun to pick it up.
7. Head back to Gimmighoul and give them the feather. You'll now be able to trade these feathers for random furniture.

Walkthrough source: Game8.