According to Donal O'Sullivan, Carolan himself probably did not use the tune as the earliest known sources for his tunes do not use the term. Petrie says that the planxty is "a harp tune of supportive and animated character, not intended for or often adaptable to words..." "it usually moves in triplets with a 6/8 measure. It differs from more ancient classes of tunes in having less rapidity of motion and is not bound to an equality in the number of bars or beats in each part." He speculates that the name may be related to Pleraca - Planxty being the English term and Pleraca being the Irish term. He says even if the term was not used by Carolan, the musical form basically originated with him.

The word planxty may have derived from the english word, "prance" or "prank". Or it may have its origins in the word "flaxaraidh" - which may be he original form o the word based on a poem about Carolan. Or it may come from the Latin "plangere" or the Irish word "planncaim" - which means to strike the harp...

There is a fair bit more in O'Sullivan. There was also a discussion on the uk.music.folk, rec.music.celtic or rec.music.folk about it. It was an excellent discussion - you might want to try searching dejanews.com for it.