Sending sons to the NW is a unique Stark decisions. You cannot abdicate like Mormont did.
Because it defeats the purpose of managing a dynasty if you can just abdicateI believe that you have to have an adult son who is not married/betrothed to anyone, and who doesn't hold any titles. Aside from that, I'm not sure.
Also, I do think that it is an odd thing that old lords with an adult son cannot abdicate to the Night's Watch, especially in the North - there seems to have been a tradition of doing so, with Jeor Mormont having definitely done so. It definitely fits the idea of the one last hunt that the Northmen have - going on a hunt in winter so that the younger generations will have more food. Of course, it could also be limited so that only men who are not married/betrothed can do so, and only if they are over 50 (or maybe 60?).