The most obvious addition to Jewish tradition is the celebration of the holiday of Chanukah, which is obviously part of Jewish law and tradition to this day.
There are also a few minor holidays mentioned in Megilat Ta'anit that came into existence in Hasmonean times (see for example 17 Sivan, which was instituted as a holiday for a victory over Migdal Tsor). These holidays are no longer observed, but they left a trace in the Talmud which argues over whether it is still in effect.
The Hasmonean ruler Yochanan the high priest, who is portrayed ambiguously in Jewish tradition, is credited (Sota 9:10) with having instituted a few takanot: He annulled the recitation of viduy ma'aser, "waking up" (reciting a verse calling for God to wake up), knocking on a sacrificial animal, hammering in Jerusalem during the middle days of the holiday, and in his time a person wasn't obligated to ask about demay before eating it.
Like Yochanan the high priest at the end of his life, Yannay (another Hasmonean king) is portrayed as having favored the Sadducees after a certain incident (Kiddushin 66a). This probably counts as changing or instituting something about keeping the Torah and commandments, but not one that this story views positively.
The court of the Hasmoneans also forbade intercourse with non-Jews (Avoda Zara 36b).