I've always been curious at what a fishing only challenge would be like and haven't tried one for myself, where you can only use items gotten from fishing and fishing crates. (or creatures that are fished up, so items from Blood Moon enemies such as the Zombie Merman, Dreadnatilus, and Duke Fishron are fair game). That's one idea. It is also possible to get shadow chest items from fishing now in 1.4.
Another type of challenge could be an ore only challenge, you can only use items that you craft from ore. That would mean most items from bosses couldn't be used. (with the exception of Demonite/Crimtane from the EoC/EoW/BoC, Luminite Bars from Moon Lord, and Hallowed Bars from Mechs)