First of all, I think Jon is their son (most people would agree) but I'm not convinced that Rhaegar and Lyanna had this tragic love affair.
Let's start with Lyanna. We know that she was very strongwilled and independent. We also know that she was very unhappy about her engagment to Robert (what girl wants to marry a drunk whoremongeror?).
So we get to the Tourney of Harrenhal. I want to remind everyone that the tourney took place in 281 AL. Lyanna was born in 267 AL, making her 14 at the time. So what we know is that she was so moved by Rhaegar's music that she began to cry and Rhaegar ended up crowning her the Queen of Love and Beauty.
Now, I think it's pretty obvious that Lyanna was the Knight of the Laughing Tree and she and Rhaegar got to know one another when he was sent to find the mystery knight. We also know that the tourney took place over 10 days and, as far as we know, Lyanna and Rhaegar didn't really have any further interaction but it was certainly enough time for a 14 year old girl to develop a crush on the Prince of Dragonstone.
Just because she had a similar personality to Arya doesn't mean she wasn't a little romantic. And when comparing Rhaegar to Robert, the choice is pretty obvious.
Perhaps after the tourney was over, they stayed in contact. Beautiful songs, secret messages, and a handsome prince - most girls would fall for that (or at least think they were in love).
About a year later, Rhaegar "abducts" Lyanna. I personally think she was on her way to Storm's End and since she was still writing to Rhaegar he came to save her from her up coming wedding.
So my hypothesis is that, like Cersei before her, Lyanna had a deep infatuation with Rhaegar but did she truly love him? You be the judge.
Now on to Rhaegar.
The driving force for Rhaegar seemed to be Prince that was Promised prophecy. Now Aegon was born in 282 so it seems likely that Elia was already pregnant at the tourney. A comet was seen over head the night he was concieved, leading to Rhaegar to believe he was the Prince.
Now some believe Rhaegar was already looking for a second wife at the tourney but it was only after Aegon was born that the maesters said Elia wouldn't be able to have more children.
That leads me to believe that he crowned Lyanna out of admiration for her skill and bravery as the Knight of the Laughing Tree. Perhaps he was attracted to her as well.
In Daenerys' vision in the House of the Undying, Rhaegar said "a dragon has three heads" so he needed to have one more child to fulfill the prophecy. Since he named his first two children Rhaeyns and Aegon, it seems he was looking for a mother for Visenya. The fact that Visenya was known to be a warrior may have reminded him of the she-wolf. A child with Lyanna would also incorperate the Blood of the First Men into the royal line - a connection to those who survived the first Long Night.
So let's say Rhaegar is pondering this over when he gets a raven from Lyanna begging him to help her escape her upcoming wedding. It seems like the perfect solution, even destined. Maybe he was infatuated with Lyanna, maybe during their time in Dorne he did grow to love her, but his main intention was the prophecy.
So that's my theory. Any thoughts?