Suppose you're an English duke living in circa 1116 and your pre-pubescent son suffers some sort of accident that renders him unable to cumulate normally or father children of his own. do I have any legal recourse to get him out of the way in favour another son who can carry on the line?

1116 is an interesting year to pick because it predates the creation of the peerage. The first Anglo-Norman baronies were feudal titles that denoted (nominal) ownership of land along with several other unremunerated duties such as defense of the realm and attendance at court. A few of the feudal barons were granted so-called honours, which meant that their land could support dozens of knights. As the building block of the feudal aristocracy, barons could also be awarded an earldom but this was an office rather than a title per se. The signatories of the Magna Carta included several earls and their dependents, notably Essex and Winchester, but all of these men were ultimately greater iterations of a baron.

Unfortunately for your theoretical duke, his creation is off by a few hundred years. The first duke in Britain is the Duke of Cornwall, which was created in 1337 as an appanage for the eldest son of the King of England in order to elevate him above the existing earls in status. But let's say that your individual is an earl with an infertile heir. If you have another male child and remain in the good graces of the king, then your eldest son will inherit your earldom and, upon his death, allow it to devolve to his youngest brother. Should he lack sound body and sound mind, he could be legally passed over in favor of his younger brother by petitioning the Sovereign although this was a risk as the Crown was leery of preserving large estates.

Should you only have daughters, the situation changes entirely. Early patents typically allowed married daughters with male children to inherit an estate in equal parts. The proceeds from the estate were then split in between the daughters' husbands and would remain in situ until one senior descendant of the co-heirs remained. While women could not vote in parliament, their husbands could sometimes exercise their rights on their behalf, making an estate a valuable dowry. This is different from the line of succession in modern peerages, virtually all of which operate upon salic law.