The entry on Lyanna gives us a lot of info:
"Eddard's only sister, Lyanna died at the end of Robert's Rebellion.
Lyanna was betrothed to Robert Baratheon, who was infatuated with her. Lyanna was not as ardent in her love for Robert, and was fully aware of his philandering ways. However, Prince Rhaegar Targaryen crowned her Queen of Love and Beauty after winning the Harrenhall tourney, passing over his wife, the Dornish princess Elia, in her favor. The exactr nature of her relationship with Rhaegar is not presently clear, but shortly therafter she and Rhaegar disappeared together, which her fiancee Robert and older brother Brandon treated as a non-consensual abduction. Their anger over the event led to Brandon;s execution by the king and Robert's Rebellion.
After the sack of King's Landing, Eddard and six companions went to retrieve Lyanna at a building Rhaegar had named the 'tower of joy'. They fought three Kingsguard knights there, including the legendary Gerold Hightower and Arthur Dayne; all were slain by Eddard and Howland Reed. After defeating her guardians, Eddard found Lyanna dying in a "bed of blood" and clutching a withered wreath of blue roses. She forced him to make a promise to her, the nature of which is unknown to the reader, but which haunted Eddard for the rest of his life in the series {Eddard reflects on what she said in this regard, on her deathbed: "Promise me, Ned";) thereafter she died. She was buried at Winterfell, in the ancestray crypts of House Stark, where Robert would still come to pay his respects, fifteen years later. {Ironically, Robert was far more faithful to her in death than she expected him to be in life.}
Lyanna was described by those who knew her as a spirited young woman, and more than a little wild; she was an excellent equestrian, and according to a roman a clef told to her nephew Bran by Meera Reed had some amount of skill with a sword as well. She was said to exhibit the ty[pical Stark coloring, and was regarded as beautifyl. Comparisons are made between her and Arya, both in appearance and temperament (a shock to Arya, who does not consider herself attractive.)"
Part of Ned's profile:
"Ned was present, along with Howland Reed, at a tower on the Dornish border known as the "tower of joy" when his sister Lyanna died there, an incident surrounded in mystery that haunted Ned ever after.
Once the conflict had ended and Robert had taken the throne, Ned returned home, bringing with him a bastard son, Joh, about whose orgins he refused to speak, even to his own wife; he quickly silenced rumors that the boy's mother was the Dornish noblewoman Ashara Dayne."
So, from all that we can ascertain:
Rhaegar "kidnapped" Lyanna, a woman he publicly named the Queen of Love and Beauty at a tournament, even over his wife which would have been customary.
Robert was sacking King's Landing, or maybe fighting Rhaegar at the Trident, while Ned and two other warriors went WAAAY south, all the way to Dorne, to rescue Lyanna.
Robert claims the throne and Ned returns to Winterfell with Jon Snow, a child whose mother he will not talk about and whose origins are simply 'he's my bastard.'
Theory: He promised Lyanna he would never let Robert learn of the child she had with Rhaegar because she loved Rhaegar and that child was the only thing left of their love. Robert would have murdered Jon, and been within his rights as king to do so, if he would have discovered. Lyannba, being a willful and spirited woman, went south with Rhaegar, and was not 'kdnapped'. With all the time it takes to wage a war, win a throne and just plain travel around such a vast landscape, it's entirely believable she got pregnant and popped out Jon in the time it took them to find where she was and get to her. She died in labor and Ned and his pals hid the whole thing in mystery because it's just a sad fucking story and they respected Ned as a warrior, a lord and a man. He can't even lie to Robert about Jon because Ned's a terrible liar and Robert grew up with Ned at the Vale (both were Jon Arryn's wards) so he'd see through honorable Ned's bad lies. So he just doesn't talk about it.
That's the theory. It's very solid, and is likely the truth.