The other two answers are correct that legally, nothing changes but practically, there's not much you can do. However, there's one effective "power" that a lame duck president has which they do not normally have: The ability to pardon without political consequences.
A president can pardon for any federal crime at any time, and no one (even their successor) can undo that. However, if they pardon someone highly controversial, it may have an effect on their chances of being reelected or their party's performance in the next Congressional election. But if they're not going to run again in the future, and there will be almost two years of other news events before the next Congressional election, they can pardon whomever they want without suffering for it, and their successor can't undo that.
For example, a third of the people Obama pardoned during his eight years in office were in those months after the 2016 elections, half them on his very last day.