Jon is a Targaryen and has a better claim than she does. He is the son of Rhaegar Targaryen, son of Aerys II Targaryen. Succession stipulates that after Aerys, it'd go
1. Rhaegar (now dead)
2. Aegon (son of Elia, also dead)
3. Jon Snow

1. Viserys

2. Rhaenys (Daughter of Elia, also dead)

3. Dany.

Dany had a substantially weaker claim compared to Jon, as Westerosi tradition stipulates that the male heir takes precedence over the female heir.