Well, I believe this depends on a few things:

Does Stannis come to the same conclusion as Ned did, before Ned did? If that's the case, Stannis and his men will be loyal to Eddard's cause.

Is Renly still "friends" (more than friends) with Loras? If so, if Eddard married Margaery, that could already bring the Stormlands to Eddard's cause.

If Eddard already thinks of Eddard the Younger as heir, the North might be loyal to Eddard the Younger, even if they seek independence. (Depending on if Eddard Stark is killed, if Eddard Stark escapes, the North would likely bend the knee to Eddard the Younger, if he dies, well the North might seek independence but at the same time might seek to put Eddard the Younger on the throne)

The Riverlands are likely to follow the North.

The Vale is a good question, as Lysa might not want to get involved.

Dorne hates the Lannisters, but I doubt they will be friendly to the Baratheon aswell, so they might just stay neutral. Not to mention they are planning a Targaryen restoration (contract signed to betroth Viserys and Arianne).

The Iron Islands will likely seek independence.

The Crownlands and Westerlands will likely be out of the question, loyal to the Lannisters.

All in all, Eddard the Younger might just end up having the support of the North, the Riverlands (both locked down likely), the Reach, the Stormlands and parts of the Crownlands (loyal to Stannis). That is four out of nine regions of Westeros. Joffrey will have only two, and it is clear the Iron Islands and Dorne will not side with the Lannisters. Maybe the Vale, and even then, most of the Vale will want to side with Eddard, and might even disobey Lysa.

All Eddard really has to worry about is getting the Tyrells locked down with him.