Patronymic Based Surnames

Surnames based on the Christian name of the father are very common in English-speaking countries.

English names terminating in "son" (or the contraction "s"), "ing", and "kin" are of this type. Thus the sons of John became Johnsons; the sons of William, Williamsons or Wilsons; the sons of Richard, Richardsons or Richards; sons of Dick might become Dickens, sons of Bartholomew might be Bartlett
In Scotland and Ireland "Mac" or "Mc" means "son of" and families which had settled in Ireland soon after the Norman Conquest have a surname beginning with "Fitz" (from the French "fils", for "son"). "Fitz", as used in England often indicates illegitimacy -- so the surname Fitzroy means the illegitimate son of the King (from the French, Fils de Roi).
The Irish "O", as in O'Brien, means the grandson of Brien. The sons of Neill might become MacNeills; the sons of Herbert, FitzHerberts; the sons of Reilly O'Reillys;

Welsh surnames can be difficult to trace since, though patronymic, they were not always hereditary. William's son Hugh, for example, was Hugh Williams; Hugh's son Richard was Richard Hughes, and so on. They also used the prefix "ap". So the sons of Thomas, ap Thomases (ap has been dropped from many names of which it was formerly a part).

There are also German, Netherlands, Scandinavian, and other European surnames of similar formation, such as

the Scandinavian names ending in "sen".
In the Slavic countries the "sky" and "ski" played the same role.

A Christian name can be altered over time. The name David, for example, has become: Davey, Davids, Dowell, Davidson, Davidge, Davie, Davies, Davis, Davison, Dayson, Davy, Davys, Daw, Dawe, Dawes, Dawkes, Dawkins, Daws, Dawson, Day, Davitt, Dowson, Dowd, Dowden, and Dowling. The baptismal name of Richard has been modified to give us: Dick, Dickens, Dickenson, Dickson, Dixon, Heacock, Hick, Hickin, Hickman, Hickmot, Hickox, Hicks, Hickson, Higgins, Higginson, Higgs, Higman, Hiscock, Hitch, Hitchcock, Hitchinson, Hitchmough, Hix, Reckett, Ricard, Rich, Richard, Richards, Riche, Richer, Richett, Richney, Richie, Richman, Rick, Rickard, Rickeard, Rickett, Ricketts, Rickman, Ricks, Rickson, Ritchie, Ritchard, and Rix.