There are two questions on EL&U (1,2) concerning names with “Mc” in them (such as McGregor), revealing that Mc comes from Mac, which is commonly translated from Gaelic as “son of”.
I have now come across the last name ApSimon. A quick Google search reveals that there are quite a few people named like that, some spelling their last name ApSimon and some Apsimon. I am wondering why there is double capitalization in some cases – is this in any way related to the “Mc” names? Is it another prefix with a similar or different meaning? In any case, the “Ap” prefix seems to be a lot less common than “Mc”.