Records point to an ancient Welsh "landed" family first using the name Badham as a surname as early as 1287. This was after a marriage to an Anglo-Norman heiress who had large landholdings. This meant joining the English "establishment", including fighting alongside Edward I at the Battle of Falkirk.
Ap Adam derives from the Welsh "mab" meaning "son" with "of" implied in the construction of the patronymic naming system used in Wales, sometimes as late as the 18th century. In this system instead of having a surname an individual was called after their father and grandfather, for example, Thomas ap Howell ap Rosser. Thomas's son William would be William ap Thomas ap Howell and so on. This surname formation from a patronymic aligns with other examples, such as ap Howell becoming Powell, ap Evan becoming Bevan, ap Rees (becoming Preece or Price), ap Richard becoming Pritchard, and ap Owen becoming Bowen. The detailed hypothesis is that "ap" becomes softened to "ab" before the initial letter "a" of Adam and the "h" crept in as a confusion with the Welsh form of Adam, i.e. Adda, where "dd" is a Welsh letter form prounounced like "th" in English.
The conversion of ap Adam to Badham becomes less than coincidental when positioned in the timeline when it occurs in documents - the turn of the first Millenium AD, a Norman invasion, a constantly "at threat" position of the Welsh Marches, and opportunities for Welsh families to improve their position and financial status by being accepted in what is now a financially rich and socially dominant Norman England.
All these hypotheses, together with the subsequent further generational integration due to migration into Britain and beyond, were the main subjects of research for the Badham One Name Society. But in practice, much time was spent helping members trace their own ancestry and helping them connect with possible living relatives (however distant!), now the aim of the Badham Family History Group.
All this and more is explored in far greater depth in Peter Badham's "Badham Delvings" book, available in the Resources menu group.
However, you can also have a look at our fascinating timeline (select from the Resources/Extras menu) that plots the Badham history in the context of wider British history. In addition you'll find many articles, telling stories about Badham families, the Badham name and Research information.