Paris has the same name as the Trojan prince in Homer's epic the Illiad, who persuades Helen to leave her husband and marry him. Is there any connection?
This is a very interesting question. I would say that the connection is by chance. There is nothing in the play that would suggest that Shakespeare was intending there to be an allusion to Homer's Iliad. In Homer's Iliad Paris is a weak man, whom Hector bails out of trouble often. Paris is also morally compromised as he "steals" Helen. Moreover, in Homer's work he actually gets Helen, even though she is married to a king.
In Shakespeare's work, the suitor, Paris, who is kinsman of the Prince not only does not get Juliet, but he does have some redeeming qualities. First, there is no doubt that he truly loves Juliet. He goes to the tomb of Juliet to mourn her death. He even sends the servant away to have privacy. This shows his sincerity. He also fights Romeo, thinking that he has come to desecrate the tomb of Juliet. Paris dies in the fray, but his final wish is to be laid next to Juliet.
The only similarity that I can see is that they are both very handsome.