Henry would imply that Martin was Henry's son. Other cultures used different ways of indicating patronymics. Over the centuries this form yielded the names Reese and Price. In Scotland and Ireland the typical patronymic form was mac X, yield names such as MacAndrew, MacDougall, MacGregor and MacLeod. Locative and toponymic bynames are another common form of byname.
Locatives are very typical of the aristocracy in England and France.
A locative byname indicates that you are from some named place. Sometimes the name of the place followed the given name directly, without a preposition, thus forming the bynames Linton, Homwode, London and York. Toponymic bynames are derived from topographic or other local features of the landscape. Names of this type are quite common in England. Eventually, of course, these usually wore down to Beech, Hill and Fen.