In Wales they used to use ab/ap as a patronym, a bit like Mac in Gallic. There might have been similar in parts of whatever they called England before the anglo-saxons came, but that’s not likely to have influenced anything by the time of Becket, or the later time when the ‘a’ was added.

I don’t think it has really survived in Wales either; the ‘a’ has often dissapeared and the p/b merged with the fathers name, like Prichard, or Bowen.