The scene we see in Season 6 Episode 3, Oathbreaker where Bran is having his flashback.
This scene is known as the Tower of Joy. This was Ned's attempt to rescue Lyanna. Howland Reed was the only other survivor of the encounter. He is still alive at this point and is the Lord of Greywater Watch. Howland will be able to support Jon's claim.
It is also possbile that Jon was legitimized by Robb and may have claim to Winterfell.
In A Storm of Swords, Robb and Catelyn have the following conversation:
“Mother.” There was a sharpness in Robb’s tone. “You forget. My father
had four sons.”
She had not forgotten; she had not wanted to look at it, yet there it
was. “A Snow is not a Stark.”
“Jon’s more a Stark than some lordlings from the Vale who have never
so much as set eyes on Winterfell.”
“Jon is a brother of the Night’s Watch, sworn to take no wife and hold
no lands. Those who take the black serve for life.”
“So do the knights of the Kingsguard. That did not stop the Lannisters
from stripping the white cloaks from Ser Barristan Selmy and Ser Boros
Blount when they had no more use for them. If I send the Watch a
hundred men in Jon’s place, I’ll wager they find some way to release
him from his vows.”
He is set on this. Catelyn knew how stubborn her son could be. “A
bastard cannot inherit.”
“Not unless he’s legitimized by a royal decree,” said Robb. “There is
more precedent for that than for releasing a Sworn Brother from his
Robb then has a council with his lords as follows:
He picked up a sheet of parchment. “One more matter. Lord Balon has
left chaos in his wake, we hope. I would not do the same. Yet I have
no son as yet, my brothers Bran and Rickon are dead, and my sister is
wed to a Lannister. I’ve thought long and hard about who might follow
me. I command you now as my true and loyal lords to fix your seals to
this document as witnesses to my decision.”
It's not clear what has happened to that sheet of parchment. Robb then sent Galbart Glover and Maege Mormont to Howland Reed at Greywater Watch, but the written messages they carried were false, in case they were captured.