In Birds of Prey: And the Fantabulous Emancipation of One Harley Quinn (2020), she changed Joker’s name to Puddin. Why did she do that?
All in all, it's just a nickname she gives him, just like some people call their loved ones "honey" or "sweetie".
Out-of-universe, there doesn't seem to be any special reason. It predates the DCEU and the comics, Harley Quinn having debuted in Batman: The Animated Series. When Dr. Harleen Quinzel became Harley Quinn, her very first line contained it:
Knock knock, puddin'! Say hello to your new, improved Harley Quinn!
The nickname just stayed afterwards, in various cartoons, comics and movies.