According to
Woordenboek van Voornamen by J. van der Schaar, Bregje is a form of
Brecht, a woman's name from a Germanic word meaning "radiant, shining, brilliant". In other words, it's a form of the same Germanic word which became "bright" in English.
Other Dutch forms of
Brecht besides Bregje are Brechje,
Brechtje, Bregina, Bregtina, and Bregtje.