Throwing only (only bombs/grenades/dynamite to mine) with all throwing weapons and for hardmode I think you can include the Daybreak, Vampire Knives and Shadowflame Knife (maybe Flying Knife) and if that's too hard use boomerangs too.
Flail-only challenge. Actually not that difficult besides struggling to get your first flail.
Spear-only challenge. More difficult than the flail challenge as there is no celestial-tier spear and the North Pole is hard to get.
Dart-only challenge. Extremely difficult. Only use dart-using weapons (blowpipe, blowgun, dart pistol/rifle)
Archmage Challenge. Only use spellbooks (eg. Water Bolt). If there is no Water Bolt then the only other available weapon is the Demon Scythe.
Tools-only. Only use weapons with a tool functionality as weapons. Pickaxes, waraxes, hammers, hamaxes count along with bombs and dynamite.
Alphaphiliac Challenge. Only use weapons (or items) starting with a single letter. So for I could see only using B, C, D, F, G, M, P, R, S and T. Feel free to expand so multi-word items count (e.g. using a Queen Spider Staff in an S challenge) or using multiple letters like W, V, X, Y and Z.
Gravity Globe. Always be flipped upside down.
Boss Drop Challenge. Only use items dropped by bosses. Start off with only a goblin battle standard and fight a goblin invasion to get a harpoon/spiky balls to progress and kill the eye. Invasions count as bosses. The player can also craft boss summoners but that it is it.
Objective for all these challenges: Kill the Moon Lord. May be nearly impossible for the tools-only and dart challenges.
Only use Broken-reforged weapons. You can reforge weapons to Broken.