(Spoilers MAIN) Do you guys think Robert offered Ned to legitimize Jon?

I think he probably did, at least once early in his reign and when Jon was still a baby.

The series states that bastards can only become legitimate heirs of their fathers by royal decree, and the idea to make Jon a legitimate Stark was even brought up twice in book three by both Rob and Stannis. So I find it hard to believe that Robert at some point didn't consider doing it for Ned while he was king.

He would have probably been as shocked as everyone else was when the honorable Ned Stark came out of the rebellion with a bastard baby, but he would have laughed it off and just made a joke about how even the coldest northerner has the occasional urge to warm themselves up with the nearest woman. He'd probably know Ned well enough to figure that said moment of weakness would still be a slight to his person, so he'd most likely offer Ned to make it go away (or at least not be as humiliating) by making Jon a legitimate Stark, even if Jon Arryn was against it (and I think he would be). Ned, however, would have turned him down, probably claiming that he should not be so easily absolved for what he did and that it would not be fair to Rob and any other children he would have with Catelyn (though of course that would not be the real reason he is against it) and after a few unsuccessful attempts Robert would just drop it since he has more important things to deal with.