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The Irish form is Ó Danachair, but was once Ó Duineachair, according to Edward MacLysaght ("Surnames of Ireland"). He gives no explanation of the name, which probably means that it can't be explained with any certainty.
It is a Patronymic denoting descent from a man named Duineachar. The second component may be the same as that in the name Fearchar, "dear man", which is the source of the surnames Farragher (Irish), and Farquhar or Farquharson (Scottish). The first component looks like duine, meaning "person", resulting in a given name meaning "dear person".
The problem with this explanation, for me anyway, is that MacLysaght doesn't give it. So it may have been considered and rejected for reasons known to Gaelic scholars, but not to me.