As of Season 7, Episode 5, *Eastwatch* the ways of being proven are adding up (as you have duly noted yourself).

 1. Howland Reed, being present at the Tower of Joy has first account of what transpired. The proof is that he is well respected noble lord and has no reason to lie about such things as he was on the opposing side of the Targaryen's during the rebellion. 
 2. Brandon Stark has "green-seer" vision in which he has ascertained this information. The proof is a bit harder here as magic is going to be looked on with a certain degree of skepticism. However, if Bran can prove he knows other things (e.g. "Chaos is a ladder) that he would no other way of knowing, people may tend to believe him.
 3. High Septon Maynard issued an annulment of Prince Rhaegar's marriage. While do not 100% know yet that this means he married Lyanna Stark, all arrows are pointing in the right direction. This is crucial as it wasn't not just any old septon, but The High Septon! This lends the religious authority to have Jon be a legitimate Targaryen. Even though the Sept of Baelor was destroyed, the Faith of the Seven is still the dominate religion in Westeros. Cersei may not want to hear it, but many lords and common folk will take the record seriously. 
 4. Jon being "dragon sensitive". I think we saw some this in S7E5 as well, when Danerys returns to Dragonstone and Jon reaches out to Drogon. This is more speculative than the rest at this point, but seems like it will pan out. It is generally accepted that you need the blood of Old Valyria (i.e. Targaryen) to have this affinity with dragons. 

All of this combined would likely convince enough of the lords of Westeros to ultimately believe in Jon's claim...