Oh it's the Finnish girl's name, like Little My from the Moomins. That is definitely an improvement, reading it as the possessive pronoun makes it look like the name of some pushy preinstalled crapware. YourSQLExperience. EnjoyData. MyCloudStorageIDidntAskFor.

Well, there is even more to it.

First of all the name My is (Finland-)Swedish, the Finnish name of the character is Pikku Myy.

The Finnish name registry contains only ~2 mentions before the publication of the hugely popular first Moomin book (and none for Myy) so this girls' name and hence the name in the MySQL really comes from the Moomin stories.

And finally, the name of Little My is a play on the Greek small letter mu, i.e. μ, and you'd be closer to the real idea and original pronunciation if you thought of it as MuSQL (μSQL) instead ;)