Last names in -ík
Q: Did my Slovak last name mean "a son of..."?
"A son of..." was not a concept that played a role as last names developed among the Slovaks and were eventually made obligatory by the Habsburgs. If anyone ended up with a family name derived from a person's first name, it was because the derived name was already quite traditional for a particular household in the village, where it would typically distinguish two households with originally identical names (see the sidebar). It all goes back to individual nicknames, not to concepts like "a son of." The latter derivation (Peterson) was the case in Scandinavia, not in the Habsburg lands.
What happened in Slovakia was that someone was customarily nicknamed, e.g., Johnnie in the distant past, and then his whole family, household, farm, manor was referred to with that nickname (Andrew of the Johnnies, Mary of the Johnnies, etc.). Then the number of the Johnnie-households (and the Pete-households, and the Mike-households...) in the village multiplied, and new nicknames gradually replaced the old ones in order to distinguish them all. The core name was sometimes kept, and a variety of endings differentiated the households.
Although it may appear almost unfathomable when viewed through English, a multitude of variations like those on Peter (see the sidebar) are perfectly regular and took place on a massive scale in Slovak. There was no "meaning" in these endings when they were employed in nicknames and eventually family names. They were merely variations of the original first name Peter. The variations were "petrified" when the last names were decreed and entered in government records.
The Slovak ending -ík in Petrík is a common derivation, meaningless in the formation of family names. It is just one of the possibile endings in this function.
If a person's first name is Peter and someone calls him Petrík, it can mean all kinds of things (the ending is not monosemantic): "little Peter," "Pete," "short Peter," "hey, Peter!," "I'm-being-dismissive-about-Peter," "dear Peter," "that guy Peter," "Peter, my buddy," etc. And the same meanings can be associated with other endings attached to Peter. None of that, however, has any bearing on how the variation became someone's registered family name.