Lady Stoneheart has almost been guaranteed not to return.
Okay, Catelyn believes the other woman to be someone named Ashara Dayne. Ashara Dayne's brother was one of the Kingsguard killed by Eddard Stark present at the Tower of Joy. Ashara Dayne was originally engaged to Eddard Stark, with Brandon engaged to Catelyn. Ashara Dayne was pregnant, although it's unknown when she fell pregnant. The child was miscarried some time before the events of the Tower of Joy (months before meaning Jon, who was born eight months before Daenerys, who herself was born exactly nine months after the Sack of King's Landing). Eddard returned House Dayne's sword (Dawn). Two days later Ashara killed herself.
So the theory goes that Ashara tipped Eddard off to her brother being with Lyanna at the Tower of Joy. Ashara may not have known about the pregnancy (if that was common knowledge Eddard wouldn't be able to suppress Jon's existence, in fact this may have meant further civil war if it had and placed House Stark in a very VERY weird position. Since there is the possibility the miscarried daughter was a Stark and specifically Eddard's, conceived prior to his marriage to Catelyn, then it paints a clear picture of why she committed suicide.
1) She miscarried the man she loved's child.
2) She miscarried a child.
3) The man she loved married another woman.
4) Said other woman had borne a child to term.
5) She told Eddard where Arthur was, resulting in his death.
The combination of these things in such a short time pushed her into making the course of action she did. Eddard himself feels guilt for all of this, and this guilt and suspicion is probably the true reason talk of Ashara Dayne is banned in Winterfell.