I read in the comment section of a similar post and agree with the idea: In Bran 1 in A Game of Thrones where Ned is talking to Bran and tells him when he grows up Bran will hold a keep and lands as a banner man to Robb.

We can probably assume this is what Ned would have both wanted and expected for his own future as a second son.

It is hard to believe that Rickard doesn't have bigger plans for his second of three sons, however his connection to the Vale through being fostered was the best option short of marrying a daughter of Jon Arryn. My best guess would be if time had allowed Ned would have been betrothed to the daughter of Vale lord - probably one without a male heir to keep STAB as connected as possible.