Here's an idea... Okay, I'll confess it's probably the most absurd concept I've posted here... but I'm gonna put it up and see if any of you can find a way to prove it to be true...
Coldhands is actually Lyanna.
Facts: Lyanna was a mess when Eddard found her. There is a tomb for her in the Winterfell crypts.
Assumption: Lyanna died. This is all from Eddard's memory. IIRC, Cat sees Ned's bones, but Ned does not remember brining Lya's bones home... he brought her home.
Facts: Lyanna was willful. Lyanna was a capable fighter. She is assumed to have been the mystery knight at Harrenhal. Benjen Stark, First Ranger, is not a personable fellow to the NW rank and file.
Assumption: Benjen, weakest wolf, survived Robert's Rebellion. What if Benjen got himself killed like Cley Cerwyn did? What if Benjen died from an illness? What if Benjen was poisoned? What if Benjen was killed by Aerys II, but was forgotten?
Now what if Eddard took Lyanna and Jon home to Winterfell? Say... Ben died and was buried in Lya's coffin. Say... Lyanna, a fighter, assumed Benjen's identity and then hightailed it to the Wall so that none of the Stark retainers would recognize her. Say... Lyanna's skills and heritage quickly elevated her to a position of authority in the NW so that none can continually pry into her life. Say... Jon is raised as Ned's *******, but his mother checks in on him every few years.
Remember the exchange between Benjen and Jon during the feast at Winterfell? Jon said, "You're not my father." To which Benjen replied, "No. More's the pity." If the words are not exact, they're close... Anyway, Benjen's pity was that she is Jon's mother and helpless to guide him. Then she decided to bring him to her... she, Benjen, put the idea of joining the NW in Jon's head.
Eddard never stopped Jon from taking the black. Why? If Jon is Rhaegar's son, then Eddard should have told him! Eddard just let Ben sway Jon and take him away... forever. Why? Eddard did not tell Jon because Jon's mother was already at the Wall and could tell him at her leisure. But Benjen disappeared soon after Jon's arrival...
Benjen is now assumed by some to be Coldhands. Coldhands is black. Lyanna is associated with the blue rose of the North. Mayhaps Coldhands' hands are not black, but dark, dark, dark, dark blue.
Edit: Also the entire Arya is just like Lyanna comparison cannot be ignored. Arya looks like Ned. Arya is willful. Arya is a fighter... will become a very proficient fighter. Lyanna was beautiful, and Arya looks like her. Arya went south and had a major incident in the Riverlands. Both spent time in Harrenhal. Arya joined the NW... mayhaps Lya did too! Both love Eddard and Jon more than anyone else in their family.