It is a misconception that the name Evans is a patronymic name. The name does refer to Evan-S, meaning son of John, however, in this case the name refers to the fact that many Welsh were late converts to Christianity and around the 3rd Century AD a huge evangelical conversion began. Converted followers took the name of Son of John (the Baptist), in reference to the John the Baptist as the baptiser of Jesus Christ and considered a cornerstone of Christian conversion. It is possible that later some did take it as being the son of their father called Evan (John), but the extended use of religious forenames being converted into surnames by the addition of "s" of Son (Jackson, Johnson etc) does not account for the huge popularity of this name in this part of the world (South Wales).
In the Welsh language the patronymic "ab Evan", results in the surname "Bevan", which is also common in Wales.