It seems to me that for a long time, the word Shlita was popularly associated with the Lubavitcher Rebbe. And in those days when I was first encountering the term and asking about it, I was told that it abbreviated SHeyichye Leorech Yomim Tovim Amein, may he live long, good days, Amen.
But lately I've heard another non-jocular explanation, something like SHeyichye LeYomim Tovim Aruchim, which would amount to the same thing without the unusual task of responding Amen to our own beracha.
What could make one explanation particularly correct? Is there a known source of this Shlita business?