A shortcut for the phrase "drop me a line" born from the routine tendency of text messaging devices to "correct" misspellings, in this case the word "line" becomes "lime".
Increasingly "drop me a line" is used more often than "give me a call", because use of text-messaging, tweeting, Facebook-ing, and emailing are becoming as or more common than using the telephone.
Because "lime" is absurdly unique when used in the phrase "drop me a lime", it is being adopted by some wanting to convey the message in a simple and short manner.
Thnx for info. Lime me.
b there at 8! Lime please.