Jon would be Daenerys' nephew.
As Rhaegar Targaryen is Dany's older brother, both children of Aerys and Rhaella Targaryen, and
Jon is Rhaegars' (possibly) illegitimate son (by Lyanna Stark), Dany is Jon's aunty.
Their family tree can be found towards the lower part of this page.
According to this recently released character association chart HBO has confirmed that Jon is Lyanna and Rhaegars' son. Also, it does confirm that (in the show at least) Rhaegar abducted Lyanna, meaning that Jon is illegitimate.
Update for Season 7:
It has been revealed that Jon is trueborn son of Rhaegar and Lyanna.
Rhaegar didn't abduct Lyanna, he loved her and she loved him back.
Rhaegar got his marriage to Elia Martell annulled and got married to
Lyanna instead. Jon's mother named him "Aegon Targaryen" on her death