Expanding on Edlothiad's answer, The War never was about Lyanna's abduction! I blame the singers for making out that it was fought as such because fighting for love is better for songs than fighting for your life.
See the timeline of these events.
Tourney of Harrenhal happened in 281 AC in which Rhaegar crowned Lyanna as Queen of Love and Beauty. When the Starks were on their way to Riverrun afterwards for Brandon's marriage, Rhaegar abducted Lyanna.
Brandon went to King's Landing to challenge Prince Rhaegar to "Come out and die". There he got himself imprisoned, the fool along with his friends Elbert Arryn, Jeffory Mallister, Kyle Royce and his squire Ethan Glover. Royal summons went to fathers of the captives, including but not exclusively, Rickard Stark.
Rickard travelled all the way to King's Landing as did fathers of the other captives. Aerys had them all killed (Except Ethan Glover).
How long do you think it took for them to travel? The captives and their fathers, all of them? Months presumably? Or at least Weeks?
Nobody raised any banners for Lyanna, nobody called their sworn swords. Everyone was just watching.
With all the prisoners executed, still no one lifted a finger. However we must note that we do not know how much time did pass between the execution of prisoners and Royal edict calling for deaths of Ned and Bobby B. It could be that they received the word from the same bird that brought the royal command for execution of the young Lords. Still, the realm did nothing in the months of prelude before the war broke out.
The factor which actually started the war was Aerys' orders to have the young Lords Robert and Eddard executed. That's when Jon Arryn called his banners. Not for rescuing Lyanna and returning her back to her betrothed, but to save the lives of his foster sons. This was the year 282 AC.
As you can see, if Robert's Rebellion was indeed built on rescuing Lyanna, why didn't it break out as soon as Lyanna was abducted? Or as soon as Brandon was arrested as the monarch didn't appear to be in the mood to legally return the abducted girl? Brave Lord Robert did nothing till then, remarkable for someone whose words are "Ours is the Fury", no Fury was seen at all. Brave Lord Eddard didn't either even though Winter had come for House Stark. Neither did the honorable Lord Jon Arryn whose words were "As High as Honor". Nor did Lord Hoster Tully who boasted "Family, Duty, Honor". The Silence of the Lions of the Rock who proudly carved "Hear me Roar" on their gates and sigil was deafening. Lord Reaver of Pyke was pretending he wasn't even there. There are your Rebels, such as they are, the Glorious Honorable Lords who supposedly fought against the grave injustice done to Poor Robert Baratheon when bad Dragonprince stole away his Wolf-love.
Robert's Rebellion was based on only one thing, which was the lives and safety of the Young Lords Stark and Baratheon. It was never about Lyanna even though Robert personally wanted Lyanna more than he wanted a crown. But since he never rose until the alternative was his death, We can assume that he wanted his life more than Lyanna.
So Robert's Rebellion wasn't based on a lie. It was based on a truth, the sole truth which is the survival instinct of every living thing. It's just that Bran has been fed the lies that it was to rescue Lyanna.