None at all. The name of the city comes from a Celtic-speaking Gaulish group that lived in the area, the [Parisii](http://en.wikipedia.org/wiki/Parisii_(Gaul)); the origin of the name of the mythological Paris is unknown. There's absolutely no reason to suspect any link.

The shared name is simply a consequence of the fact that there are lots and lots of names in the world, and only a limited set of phonemes to go round. Sometimes you get overlaps. Chile has nothing to do with chilli, Turkey has no connection with turkeys, Manchester has nothing to do with the masculine gender, and Paris has nothing to do with Paris.