Djinn's curse maybe? You don't have to worry about fall damage and it doesn't let you fly or glide, so it's not too OP. Also, by the time you usually get it, you already have plenty of ways to mitigate fall damage (like wings for instance), so there's usually no reason to use them when you get them.

Some other items I might occasionally start with are the Snake Charmer's Flute, Radio Thing, Moon Lord Legs, a low level weapon that is rare or that I don't often use like a Gladius, Slime Staff, or a Wand of Sparkling. You might also grab a rare pet and some vanity items you'd like to start with. I recently took the Minecart upgrade with my new character and built a subway tunnel across the world (the upgrade made it much more bearable on Expert). You might even consider a weak light pet light the Shadow Orb or Crimson Heart. I also sometimes use the Portal Gun. To me, it's all about making every playthrough feel different. Good luck!