Even if Bran/when Rickon resurfaces, nullifying Jon's claim to Winterfell (which he would never take from one of his true born brothers), in order for Jon to have a claim in the first place Robb must legitimise him as a Stark. So either:
i) Neither Rickon (eaten on Skagos) nor Bran (too far north of North) show up and Jon becomes the Stark in Winterfell and rallying point for the northern lords,
ii) Either Rickon or Bran show up and become the Stark in Winterfell and rallying point for the northern lords.
Either way, for the will to be important at all, Jon Snow must become Jon Stark.
Firstly, although the idea of the kingdom of the North is still clearly alive and well in the minds of some (Manderlys, Mormonts etc) many of it's strongest supporters are dead or imprisoned. There doesn't seem to be anything left to 're-create' the kingdom as it were, with manpower and support massively reduced during the war. If anything the Riverlands territories are in an even worse state.
I'm not too convinced that the remainder of the North still wants sovereignty as much as they want to get back to being the North-under-Winterfell.
The sheer devotion most of them have to avenging the Starks tells me they just want to get them back in some fashion, rather than a KitN per say.
Secondly, the will was made with the assumption that Bran and Rickon were dead. If Rickon shows up is Jon still the heir?
I highly doubt even a fully legally legitimate Jon Stark would be allowed by the general Northern community, and much less actually want, to "usurp" the claim of either Bran or Rickon. If either of them are Lord I see Jon as castellan, or as Hand if they become KitN.
Thirdly, Jon has already refused an offer of legitimisation and the North from Stannis, will the will really change his mind?
Legitimisation from his "brother" and (if i may say) true king Robb would probably hold more sway with Jon than the same offer from the leader of an invading army (admittedly a helpful one), partially because there would be infinitely more closure attached and also because he wouldn't have to burn down the Godswood.
Admittedly, this is assuming that one of the three of them actually make it to a non-Bolten infested Winterfell, before the Others kill everyone, and Robb's will makes it out of the swamp. I will also make no apologies for being consumed by the GNC at this point, which probably coloured my opinions!!
Edited by Friendly Thenn, 20 June 2014 - 09:26 AM.