Maxson, the son of Roger Maxson and the father of John Maxson, was the second High Elder of the Brotherhood of Steel. His first name is not mentioned in any sources.
Background
He was only a teenager in 2077, during the Exodus of the Mariposa Military Base soldiers to the Lost Hills bunker. His mother died during the Exodus. In 2135, Roger Maxson died of cancer, and his son took over as the High Elder of the Brotherhood.
In 2155, The Brotherhood sent out a few squads of scouts to track down a raider gang known as the Vipers - it was more of a training exercise conducted by the High Elder, as the Brotherhood was convinced that small detachment of troops in power armor would be sufficient to deal with a group of raiders, no matter how large. One Brotherhood squad, led by Maxson, found the Vipers. Expecting the raiders to break and run, Maxson didn't take into account the religious ferocity of the Vipers (or their poisoned weapons), and when a single arrow nicked him with his helmet off, he died within hours. John Maxson, his son, took up the role of Elder, and Rhombus became the new head of the Paladins.
Notes
In the year of his death in 2155, his minimum possible age would have been 91. His participation in a military engagement in his 90s does not make much sense. This could be reasoned as him simply wanting one last fight before he succumbs to age, or the power armor having the ability to give an old man a stamina boost.
Appearances
Maxson II does not appear in any game, as he died before the first Fallout game. He is mentioned in a terminal entry at the Citadel in Fallout 3 though.