By "last resort", I mean: a course of action or option that one would only use when all others have failed, because it's unwanted. I can't think of any reasonable way to translate this into Esperanto. I feel like simply saying "fina elekto" doesn't express that it would only be considered out of necessity, and isn't simply the last choice in a list of choices.
The answers to this question so far point out an important issue in translating expressions like this that are used for a wide range of nuanced meanings. The semantic fields, that is the range of meanings, of the expressions in the two languages differ.
It's generally safe to use a fairly neutral translation like la lasta rimedo most of the time. I'd use it if I were translating something like, "As a last resort, we can always pick up one of the pre-made dinners from the grocery store." There's no emotional connotation to it in this case. It also has the advantage of being found in at least one dictionary, The Comprehensive English-Esperanto Dictionary by Peter Benson. English Phrases and Expressions in Esperanto by Felix Woolf supplies fina rimedo and lasta instanco as well. To my ear, rimedo is a better translation of resort than instanco, but I think they'd both be understood.
However, if I were translating a text about the Cold War which referred to the doctrine of Mutually Assured Destruction as the last resort, I'd go with something that at least implied some desperation, like Andrew's suggestions of malplejfarindaĵo, rifuĝsolvo or even konsilo koŝmareca.
La lasta rimedo seems like the best bet in my mind.
Some good answers. What about la lasta ebleco or la finfina ebleco?
Other possibilities are la finfina rimedo, la lastranga rimedo, la lasta fareblo, la rifuĝsolvo, la malespera solvo, la konsilo koŝmareca, la finsolvo de neceso, la malplejfarindaĵo.
"As a last resort" = lastarimede, lastasolve, finfinrimede, rifuĝsolve, malplejfarinde etc