Q: Does the Rule Mystery Play require one to play the specific card flipped up from the top of the deck?

My friends think you can add it to you hand, and play some other card from their hand.

A: You are correct, your friends are incorrect. You pull the top card off the deck, and immediately play that card. You do not get to add it to your hand, or play any other card from your hand.

Analogues of Mystery Play include (some with slight variations such as conditional requirements for use):
Wormhole in Star Fluxx
Magic Shoes in Oz Fluxx
Egads! in Batman Fluxx
Open The Door in Monster Fluxx
Magic Portal in Adventure Time Fluxx
Chemical X in Cartoon Network Fluxx
Time Doorway in Regular Show Fluxx
Shiny! in Firefly Fluxx
Great Idea! in Stoner Fluxx