As Barrie notes, dilemma is not only spelling truest to the etymology, it's the only one attested to in any major dictionary, and it is by far the most common.

As to the cause, the aberrant dilemna is almost certainly hypercorrection; if common words like solemn, hymn, or autumn brand a silent n, then surely this Greek philosophical term would as well. Like pluralizing octopus as octopi, saying between you and I, or pronouncing habanero with an ñ, the spelling is then perpetuated by well-meaning but mis-remembering teachers, editors, and so on.


As far as I’m concerned it’s dilemma. The Oxford English Dictionary has just one citation for dilemna, and it’s from 1551. The double m spelling is consistent with the word’s etymology. It’s from Greek δίλημμα, by way of Latin dilemma.