Is there documentation to prove that the Aramaic word "דילמא" is a transliteration or derivation of the Greek word "dilemma"?
If so, this fact is informative to G'mara-learning in understanding the precise connotations of terms.
From Jastrow, page 299:
This seems like a better, and more straightforward, etymology. There is a clear basis in Ezra 7:23: כג כָּל-דִּי, מִן-טַעַם אֱלָהּ שְׁמַיָּא, יִתְעֲבֵד אַדְרַזְדָּא, לְבֵית אֱלָהּ שְׁמַיָּא: דִּי-לְמָה לֶהֱוֵא קְצַף, עַל-מַלְכוּת מַלְכָּא וּבְנוֹהִי. 23 Whatsoever is commanded by the God of heaven, let it be done exactly for the house of the God of heaven; for why should there be wrath against the realm of the king and his sons?
where it is clear that di+le+mah is 'that + for + what'. And also the parallel to the Hebrew cognate shema = she + la + ma. Of course, words shift slightly in their meaning with use.
I must compliment you on your good eye in spotting the Greek word with similar meaning. It might be useful as a mnemonic, as often dilma can be used to speak about two possibilities.
But ultimately, I am fairly certain that it is a false cognate: