No one wanted to take Maverick’s teacup. He is sharing his story so the same doesn’t happen to you.
Completing the Teacup Quest
The person you need to return the Teacup to is Alice. If you find it on your first run through the story, however, you can’t give it to her yet. This led me to fruitlessly look for the Mad Hatter or someone else who would take it off my hands.
So then, how to solve this quest? Well, not much further along in the story, you will get the option to revisit old worlds and complete sidequests there. When you return to Alice at that point, she will accept the Teacup and give you a new quest.
For the other sidequests you get access to at the same time, you’ll need to find hidden backdoors. To be able to do that, you’ll need to upgrade your Scan ability first.