No specific person made up the lie. If anyone is to blame, it's Rhaegar and/or Lyanna. They ran away, even though he was married and she betrothed, and from what we know they told no one but the Kingsguards.
This is a deeply irresponsible behaviour and let to a massive catastrophe in the form of a civil war and the deaths of not only themselves, but also their family members and thousands upon thousands of soldiers, civilians, knights and lords. Rhaegars idiocy is unbelievable. How is it possible that a man loved by everyone (except Robert) causes all this and then doesn't even try to stop it by telling the truth?
And Lyanna. I almost can't believe she chose lust and love before her family. Her fiancé? Sure - she didn't chose him, but still we have no indication that she hated him ("Love is sweet, dear Ned, but it cannot change a man's nature" is all we've read her say about Robert, and that does not seem hateful). Although she must have heard of her father's and brother's deaths, and that her other brother and fiancé were marching to war, and that her lover and husband was going to do his best to kill Robert and route Ned's army, and even if he didn't intend on killing him ...marching on someone with 35 000 men means there's a high probability that he dies.
Lyanna Stark chose lust and love above her own family's lives. Is that what is meant by "willful"? She also knew that Robert loved her, and that he had no reason to suspect she ran off willingly. Likely Robert wouldn't have believed the raven scroll saying she loved Rhaegar and was going to marry him, but isn't one failed raven worth the shot at stopping a massive war where her family tries to kill his family and vice versa?
Also ... Rickard tried to marry Lyanna off for southron ambitions. The lord of Storm's End and the liege lord of all the Stormlands is a dear, dear price ... but he would have been even happier having his grandkid sit the Iron Throne. Surely both he and Brandon would accept it - Ned definitely would.
There's still the problem with Aerys though, and since the war wasn't based on a lie but on murder and conspiracy to commit murder, it's likely it would have happened anyway - the rebels simply had no choice. But Lyanna did, and there is not a single evidence suggesting she tried to stop the disastrous upcoming events.
I'm not bashing Rhaegar as much because I have no hope in him being a good man. Lyanna's actions bothers me though.