Nobody knows for certain. In Greek the name is Hómēros, and as a common noun the word meant "hostage". The ancient biographies of the poet, such as the Life of Homer that tradition attributed to Herodotus, give another explanation: in the language of the people of Cyme, in Aeolis, hómēros meant "blind", and the nickname would have come from the poet's blindness.
These biographies also say that his birth name was a different one, Melesigenes, "born by the river Meles", the river of Smyrna. They are late legends, which interpret the name from the image of the blind singer, already present in the Odyssey itself in the figure of the bard Demodocus.
Some modern scholars have proposed an etymology tied to the craft: hómēros as "the one who joins" or "fits together" the song. None of the explanations is certain, and the uncertainty of the name goes with the uncertainty about the man himself.
On Clavis: The Iliad, The Odyssey.