Hm, an interesting potential TL, with a lot of potential and fascinating consequences. If you go with a single POD, that of the birth of a son to Henry I and Adeliza of Louvain, then one would/should focus on what ripples out from that. Henry and Adeliza were married in January 1121 (thank you Wikipedia), so I'd say that the absolute earliest that the boy would be born would be at the end of the year, so sometime in December 1121. As a result of that, Henry would focus on making the boy (Richard/Godfrey/Henry/William/whatever) the focus of the succession, and everything he'd do from that point on would be geared toward the goal of providing his son with the most stable realm possible to inherit.
In regards to Matilda, I don't see the birth of a half-brother influencing her fertility and that of her first husband, so I'm thinking that they would still be childless and he would die on time in 1125. With his death and her own lack of children, Matilda was limited in her choices - entering a nunnery, remarriage within the HRE, or returning to her family in England. I can't see someone as active and forceful as Matilda choosing to take the veil, so that option is out. Remarrying is something that would probably happen whether she stayed in the HRE or went back to England, but she might want to go where she would have the most influence. If she remarries some random German prince, she'd just be one among many princesses, whereas if she goes back to England, she'd be a big fish in a smaller pond. What's more, her father might want her to come home, seeing her as a potential pawn in preserving the realm for her half-brother. Matilda might even think she's a potential candidate for the regency should her father die while her brother is still a minor.
As for Henry himself, the tradition is that he died from eating a "surfeit of lampreys" against his physician's advice. If Henry has a second son, I could see him possibly being a little more cautious in his own health, perhaps being determined to live as long as possible to give his son as much time as possible to grow up and learn, so maybe he actually does take his physician's advice and limits himself on how many lampreys he eats? That could theoretically extend his life a little.
Henry would also probably focus on rebuilding the alliance with Fulk of Anjou, which had fallen apart over their squabbling over Fulk's daughter's dowry (she had died on the White Ship as well), so Matilda may very well still end up married to Geoffrey of Anjou, though Henry might offer up a cash dowry of some kind as opposed to lands/castles in Normandy. Henry didn't want to give Geoffrey a foothold in Normandy, and so kept things muddied as to when the new couple would take control of those castles in TOTL, so he might just avoid that all-together by providing offering a generous sum of cash or something. If the marriage still happens, we'd theoretically still have Henry FitzEmpress (OTL's Henry II of England), which could lead for interesting times in the future.
If Henry does die while his son is still young, I definitely think it's possible that there will still be some measure of a shake-up. Stephen and his brother, who was the Bishop of Winchester, might still seek to seize power, but I suppose it also might depend on where the boy is when his father dies. IOTL, Matilda and Geoffrey were in Anjou, supporting rebels against the royal armies, which rather awkwardly contained some of her allies. Still, in this case, if the boy is in England or had been sent back to England the moment it appeared that his father's illness appeared mortal, Stephen might face a tougher challenge. IOTL people were wary of accepting Matilda as Henry's heir, both because it was such an irregular move and because Matilda's behavior had not won her many friends. Many of those that supported her seemed to do so only because they felt bound to uphold the oaths they made to Henry I. ITTL, we have a male heir who has been touted as such pretty much from birth, and likely entertains a wider support base than Matilda ever did. People like Robert of Gloucester (who supported Matilda's claim until his death in 1147) would be more likely to rally to the late king's son than to a nephew, no matter how influential he was (and Stephen had influence).
Instead of going for the kingship directly like he did IOTL, I think it's possible that Stephen might have tried to take over the regency for his cousin, perhaps pushing aside others who could or would also be part of the regency, like Robert of Gloucester, Queen Adeliza, maybe Matilda (depending on whether she and her husband were at odds with Henry before his death). His doing so would likely alienate a lot of people, and possibly cause a fair bit of upheaval in England and Normandy. And that's not even covering what rivals like France would do in the face of all of this.