The clan didn't had to be related, it could might as well just be a kind of bonding between some different families - for example organized some originally Lithuanian (noble-)families themselves into a 'clan' with the coat-of-arms Tra,by, when they moved to Poland in the first period of the Jagiellonian dynasty. Families in this 'clan' were my own Jaszczuld/Jaszczold (later Jaszczolt), but also the famous Radziwill-family.
The families attached to these unique Polish group-coat-of-arms are called herbowni - herbowna in singular, as I recall.
You can use that when you search for more information about its structure.
PS: This site: herbarz.neostrada.pl/str/rody/f.htm gives a good overview of many Polish szlachta-families relation to different coat-of-arms, but yours isn't there. But I would just think of it as a name meaning some like 'son of Francis', so it's not that uncommon in my ears.
I think to have read that those -czyk names were most common in the Ruthenian-borderlands, around the Kresy Wschodnie.
well, if you are really interested in your family's coat of arms, the internet websites which do searches on this are just the starting point.
they don't have everything on them. the Polish zlachty was actually quite numerous compared with the gentry in other countries. and there's a lot of detailed history about their use of coats of arms which has not been assimilated by the websites which deal with this subject.
there are extremely lengthy, detailed tomes on the subject of polish heraldry by authors such as Zarnickiego which may have information about your ancestor's use of coats of arms which you simply won't find on an internet site.
if you want to be sure of getting the right answer, i recommend you somehow get in touch with a polish researcher who can go back to these kinds of original sources for you...
try heraldic site, wikipwedia pl , and herb search , and society of polish nobilty, my family shield is KORAB, awarded in approx. 1424 to an ancestor. first mention of this was in the late 1200's amazing