This might be a bit iffy, but hear me out:
We all know of the White Ship disaster and the turmoil that followed Henry I's death. As well as the triumph of Henry (II) FitzEmpress.
I came across something interesting, namely that Edgar (II) Aetheling, son of Edward the Exile and grandson of Edmund II Ironside, was reportedly still living at the grand old age of 110 according to the Northumberland Pipe Rolls of 1158 and 1167. The general way this is explained is similar to Tom Parr's 169 years of age, that it was simply someone (a son or grandson) with the same name.
Enter the realm of fiction: in Scott's Ivanhoe the title character is in love with Rowena, although Rowena's warder intends to marry her to another Anglo-Saxon noble in order to strengthen her claim to the blood line of (I think) Athelstane (I think there's a part about them throwing the Normans out, too). The marriage doesn't happen, though.
And this got me wondering. Say Edgar has a son, named Edgar Aetheling II around 1070s/1080s, and either this son, or a grandson (Edgar Aetheling III) born around contemporaneously with Empress Matilda, marries her instead of her getting hitched to Geoffrey of Anjou. It's a stretch, I'll admit, but if there was a homegrown royal candidate, like Edgar, could/would Henry I agree to the marriage? What would the repercussions be of the match?
PS: I realize it might take some butterfly herding (although, maybe not as much as might be thought, especially if Edgar Aetheling did have a son and/or grandson with the same name that he was still thought to be kicking around England halfway through the 12th century.