Depending on the variant, I've found quite varied meanings.
For Pełczyk: from pułk = regiment (formerly połek, połk). For Pelczek (+similar spellings with l): from German proper name Pelz, which in turn comes from the name Balthasar or from Pelz = kożuch = sheepskin coat. For Pilczak: from pilić (popędzać, ponaglać) = to urge, to rush or from pilny = urgent or from piła = saw. For Pilsak: from pilśń = felt or from pilśniak = a bag made of felt.