The Prince of Dorne in the House Martell is Doran. So it could be that “Dorne” is derived from an old version of “Doran.” We know the name “Daranis” didn’t catch on, so it’s safe to say that the House Martell won out over other candidates like Dragoris or Draharis. And even if they did go with those names initially, they would have to change after being invaded by King Mors Martell and his army and later became Dornish once again under Yandel flags and banners. I think we can definitively assume that this is how Dorne got its name!