According to the Free Dictionary, dropping someone a line means sending them a short message.
Is this correct? I always thought it meant phoning someone, the line referring to a telephone line.
It refers to a line of text in a letter, so it means sending a short message (maybe with just one single line of text).
(If you drop the line during a phone call, that means you're hanging up)
History says the telephone was invented in 1876. I imagine A. G. Bell would be amazed at this Ngram (could he see it) if, indeed, it refers to telephone communication!
Drop me a line invites someone to write something to you: a message, a note or a short letter. It probably has to do with the lines that comprise a piece of writing.
I think you are confusing the line in "Hold the line" with the line in "a line of text". The statement you posted has the second meaning.
"Drop a line" also means to install a phone line, and the meaning has portmanteau'd into "give me a call" in many circles. This I can verify from first-hand experience.
I think the confusion here is in the meaning behind 'a short message' in freedictionary.com.
A 'message' in this case does not necessarily mean a text message. It can just as easily mean a phone message, and is very frequently used to mean just that.
It's funny how many phrases in language completely lose context with their origins over time. "Drop a call" seems like a different nuance than "drop a line". And there is the action of dropping a line as in the actual telephone cable line being dropped into the ground for installation.
I also interpret "drop me a line" to come from dropping someone a line of rope to reach them (connect with them). By that interpretation, and mode of communication involving any kind of line seems to qualify – a telephone line, a line of text, or even a pickup line in a bar (which has the added irony of dropping vs. picking up) :)
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