Does a stringency become a custom? For example, refraining from eating gebrokts on Passover

Certainly, when a person consistently observes a particular stringency, it becomes a custom—unless from the outset he thought it was required by strict law and not merely a stringency, or unless he initially stated that it was without a vow.

This also applies to the custom you mentioned of not eating gebrokts on Passover, observed especially among Chassidim. They are careful not to eat matzah that has come into contact with water even after it has been baked; they do not moisten the matzah or eat kneidlach in soup, and so on. Anyone who has followed this practice must continue the custom of his ancestors. If, however, it is difficult for him and he wishes to discontinue the custom, there is no prohibition, since it is permitted according to strict law; nevertheless, he must have his vows annulled.

Indeed, regarding Sephardim who have accepted the rulings of Maran, the Shulchan Aruch, Rabbi Ovadia Yosef of blessed memory wrote that even if a family observed several stringencies on Passover while still in the Diaspora and continued doing so in Israel, it is more proper for the children, after leaving their parents’ home, to follow the view of Maran. Ideally, they should have their vows annulled; in any event, they are not obligated to continue their parents’ custom after immigrating to Israel.