I exclaimed out loud when I read about MySQL. I never liked the name before, and now I do.
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 ;)
Roughly speaking, the <y> here is close to what is traditionally transcribed in English as [i:], except rounded, i.e. [y:].
Many caveats apply, this probably being Finnish Swedish (which should have the same or a similar vowel), my Swedish being rather rudimentary, etc.