A dilemma is “a situation necessitating a choice between two equal, esp. equally undesirable, alternatives”, or “a problem that seems incapable of a solution” [source].
It comes, via Late Latin, from the Ancient Greek δίλημμα (dílēmma, – ambiguous proposition), from δι- (di-, having two of) and λῆμμα (lêmma, – premise, proposition) [source].
Today I spotted the word trilemma in an article in The Spectator. I hadn’t seen it before, but from the context it appears to be a variant of dilemma involving three choices.
According to Wiktionary, a trilemma is “A circumstance in which a choice must be made between three options that seem equally undesirable” or “put another way, in which a choice must be made among three desirable options, only two of which are possible at the same time.”
I thought trilemma was a recently-coined word, but according to Wikpedia, it was first used in writing back in 1672.
Probably trilemma has been coined several times independently; the idea is obvious, but not needed often, so the word doesn’t spread much once it’s coined.
Plus, “dilemma” is normally used in a generic sense to mean any sort of difficult choice, when the options are undesirable or contradictory, without reference to the particular number of choices. For most uses, “trilemma” is simply not needed; few people have heard of “trilemma” (limiting the audience) and “dilemma” is more than adequate as a substitute.