A while back I asked what if Richard I Lionheart didn't die in Normandy, how this would affect the signing of the Magna Carta [
www.alternatehistory.com/discussion/showthread.php?t=123199]
Let's turn this around a bit. Let us suppose Richard's attempt to unseat his father Henry II (with the help of his brother John Lackland and Philip II of France) fails. Richard is forced into exile, John goes with him.
I'm not sure which rebellion of Richard's you are referring to; I'm assuming the last one, in 1189, since that actually ended with some measure of success. Given that Henry was dead within a few days of this, it sort of depends on what you think caused his death. He certainly wasn't in terribly good health at the time, but Hoveden implies that Richard might have been responsible. Either way, well, John being exiled seems a bit much. Off the top of my head I can't remember if he took a side in the rebellion, but he always was much closer to Henry than Richard and the rest.
Also, while I'm not sure if it still applies in Henry II's time, it does bear mentioning that bastardy was not as stigmatized for the Normans as it was for other European nobility (William the Conqueror springs to mind). Henry II's daughters might be an issue. However, since by this time they're all married and well away from England, it is certainly possible that the nobility might prefer an English bastard to a daughter married to a foreign lord.
It does not hurt that the presumptive new successor (Geoffrey, who was Henry II's chancellor at the time) was unwed. Promising to marry him into one of the prominent English families could probably win more support for legitimization.
I'm not sure how viable an option this is at this point in time. Think back to the start of the Anarchy. Alongside Stephen, the strongest landholder in England is probably Robert, the Earl of Gloucester. He was Henry's favorite bastard, but never made an active play for the throne. And this was in large part because, as an illegitimate offspring, he could not inherit property and would have faced substantial opposition from the church. Throughout all of the chaos of the Anarchy, despite being one of the primary powers in the realm, despite rebelling against Stephen and later becoming the leading supporter of Matilda, Robert never attempted to seat himself upon the throne. This is telling. Using another of Henry II's illegitimate sons as another example, Morgan was named bishop of Durham, but the pope would not consecrate him unless he declared himself to be a legitimate child.
Now, obviously, Henry can (try to) do as he wishes while he is alive, but sooner or later the issue of succession is coming up. And there aren't too many options out there. There is Arthur, who is a small child. There is Richard, sulking around in exile and probably plotting his return. There is John, and god knows what he is up to. Then you have Geoffrey Bishop of Lincoln and William Longespée floating around, and god knows what they do.
My guess is that things end up rather similar to the Anarchy, actually. Assuming that Henry and John are not reconciled, then Henry names Arthur Duke of Brittany his future heir, while in the meantime leaning heavily on his two foremost illegitimate offspring. Once henry dies, assuming that the bastards don't try to make a power play, you have one or both of them backing Arthur while attempting to play kingmaker, and you have Richard return from exile, with Philip's backing, trying to take the throne. It's going to be messy.
