Well, I'm already in the middle of making a hold on the theme of "
what if KDD had rock golems?"
so that's a strong contender for the first question. They are extremely versatile as puzzle elements and enforcement mechanisms, with their non-sliding movement and making rock piles when killed (although with AE elements alone, there's no way to move or remove rock piles, two facets that will give them even more puzzle potential).
The only other JtRH candidates I'd consider would be Halph and the Slayer -- both elements that have a lot of potential uses; but the disadvantage is that to make them as versatile as they are, they have to be highly complex, whereas rock golems have an extremely simple ruleset that can be fully learned in a minute.
From TCB, in chat I suggested adders as a tool for dealing with wubba hordes
but a more serious suggestion would be the speed potion, which is a complete game-changer and gives the player a lot more power when manipulating monsters. It's surely not an accident that in between the releases of JtRH and TCB there were at least six Feature Requests threads requesting the speed potion!
From TSS... my gut instinct answer is either gentryii because they're so cute, or time tokens because they have absolutely huge puzzle potential. That said, I find time token puzzles to be a bit hit-or-miss because they can get so confusing that it's easy to get really stuck and have no idea what a room is asking of me. So after more thought, I'll go with the ability to change weapon, which is both another complete game-changer
and simple enough to be comfortable to play with.
____________________________
50th Skywatcher