The aspect which probably interests LDS about Elijah above all others, is, I think, the keys of the priesthood.
The question is, are there any echoes of this in Jewish tradition?
The answer happens to be a yes.
Rabbi Yochanan says four keys has the Holy One, Blessed be He not handed over to any creature in the world, and these are they, the key of rain, the key of prosperity, the key of the graves and the key of fertility (as in child-bearing), but when they were needed he has given them to the righteous. They key of rain to Elijah, as it is said (1 Kings 17:1 "there shall not be dew nor rain these years, but according to my word.