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”.
"Ap" means "son". The Welsh used to use a patronymic system, but surnames eventually fixed after Welsh union with England. Iceland continues to use a patronymic system.
The Welsh patronymic system describes family trees in terms of the male line only and records the family association in the 'ap' or 'ab' prefix (ap is a contraction of the Welsh word mab, which means son). So, Rhys ap Dafydd means, in English, Rhys son of David.
Modern Welsh surnames such as Powell, Price and Prichard are the result of this contraction and a progressive tendency to Anglicise Welsh names: under the patronymic system they would have been ap Hywel; ap Rhys and ap Richard. The names Bowen and Bevan were derived in the same way.
Apsimon/ApSimon/ap Simon are possible variants; my paternal Great grandfather was Simon Jones of Bala (d.1873); his son Thomas Jones, b. 1843, my paternal grandfather, made an entry in the family bible in 1855, signing as Thomas Apsimon; this form is used in obituary notices for his father. This is the earliest occurrence as a formal surname that I have come across, though use as a descriptive term 'X, son of Simon' occurs earlier in documents. The use of the capital S is an attempt to make the name readable to English speakers.