Eddard's original love interest, Ashara Dayne who killed herself by jumping out a tower window back around the time of Robert's rebellion. In the first book there is a conversation with Catelyn where she asks him about the rumours. He does not deny it but refuses to ever talk about her again.
Second possibility is Wylla (a wet nurse in service to House Dayne). This is who he told Robert was Jon's father, but he may have been misleading Robert in a roundabout way on purpose to hide the truth since that is the person who Robert was thinking of when he was asking the question.
Third of course is Rhaegar and Lyanna. This would explain why he had to hide Jon's parentage from Robert and why he is so haunted by his sister's dying words of "Promise me Ned..."
Robert hated the Tagaeryn's so much and he probably would have demanded Jon Snow be killed, thus why Ned has to pretend he is his father.
Great article about it here (taken from observations during the first three books)
Past Events: Jon Snow's Parents ~ Tower of the Hand