How do the "choose one" chests work?

There are 4 "Pick One" chests, which show you up to 3 items from the same set of 4 between them

As you've probably noticed, Cadence of Hyrule doesn't have any restrictions on the order you do the dungeons in - each of them will contain exactly one of the "Pick One" chests.

No matter the order you find them in, the first will display 3 of the 4 items, while subsequent ones will display all the remaining items (depending on how many are left). To use your example, if you'd picked the Cane of Somalia for the third chest, you'd find the Poisoned Arrows in the last one instead. (This also means the third item in the second chest you'd forgotten was the Poisoned Arrows.)

There's no "wrong" choice, you will get all 4 of the items eventually.