*POTENTIAL SPOILERS*
As DDFirst Name Dd stated, I believe that <PERSON> stated that <PERSON> Grand-Aunt married into another house, which I believe is house <PERSON>, although I may be wrong about this. If for whatever reason these distant relatives are ineligible to inherit, such as dying without issue, then there would likely be another house who can trace descent through a grandparent or great grandparent to a direct male-line ancestor of <PERSON>. Unfortunately, it would seem that Westeros, or at least the north, does not have a directly clear system for dealing with succession, and as such the claimant with the most direct blood relation to the current <PERSON>'s of Winterfell may not be the best potential claimant. In Westeros there are numerous examples of claimants with lesser claims being favored over the most direct heir, such as the first Lord Paramount of the Reach, who was granted dominion over Highgarden and the Reach despite the fact that while numerous Reachman houses claimed direct descent to the Gardeners through the male line, such as the houses of Oakheart, Redwyne, Fossoway, Oakenshield, Tarly, and many others. 725
This is because in Westerosi culture the ultimate authority is the King. If a potential claimant has no real chance of holding said title, it would be unlikely that the current king would appoint another person to said title. As <PERSON> is the King in the North, so long as that kingdom continues to exist, his designated heir, <PERSON> in the books, if the will of <PERSON> had been read, and only if <PERSON> was willing to forsake his vows, and the other northern houses were willing to accept his as their king, then he would become the new Lord of Winterfell. However practicality would likely dictate that another, such as Lord <PERSON>, or possibly even a member of house <PERSON>, if they are in fact the house that <PERSON> mentions, would step up to claim Winterfell. In the end, the heir to Winterfell is whoever is able to hold it, be it a potential husband to <PERSON> or <PERSON>, a distant cousin backed by the other northern houses, or even the <PERSON>, any could claim legitimacy, and in the end, sucession in Westeros is not about who has the best claim, but who has the most power. 725