Patronymics is a naming convention in which a child's name is derived from the combination of the child's given name (first name) with their father's first name in order to create a complete name.
This naming convention is very different from present day naming conventions in which a child's last name is typically the same as the father's last name. The patronymics naming convention has been in use in a variety of English, Welsh, Spanish, Slavic, and Scandinavian countries.
Let's say we had an English family in which the father's name was "John Davis". Let also say that he and his wife had a son. Now, if this family were using the patronymics naming convention, Due to patronymics, John Davis's son "Evan" would be named "Evan son of John" or shortened to "Evan John".
This is a very simplified example of the name and language variations related to patronymics, but this example illustrates how patronymics works as a naming convention. You can also see what the challenges are as it relates to family research. Basically, if you were searching for Evan Johns father, you would not find him by searching on Evan John's last name. Instead, you would need to search for a variation of his first name, John.
In parts of England and most of Wales, patronymics was a culturally accepted naming convention up until the 16th and 17th centuries, although you will still find some use of it in Wales today. The English had actually embraced the more familiar last name, naming convention earlier than the 16th and 17th centuries.
In Wales though, before the "Act of Union" of 1536, most Welsh families used patronymics as the conventional method for naming children. After that date, Welsh culture slowly melded into the English culture and, as a result, the Welsh began to adopt the more "anglicized" (English) approach to naming children. But this did not happen overnight and it took the Welsh a very long time to accept and adopt this new anglicized naming convention.
The important point here is that if you are searching for a Jones ancestor born in the UK earlier than the mid 19th century, be mindful that patronymics could be a factor. You may find that their father did not have the same last name.
For more information about patronymics in Wales, check out an interesting book titled The Surnames of Wales by John and Sheila Rowlands. They go into greater detail about patronymics related to Welsh naming conventions.