I know the name is the Queen's childhood nick name, lots of us do that, I've certainly got a nick names derived from actual names for my grandchildren as I did with my children when they were little ...........but in reality they are ridiculous, if we are out in public I have to revert to actual name. Lets face it such names are terms of endearment used within the family and to an outsider sound twee, and that's exactly how Lilibet sounds to my ears. However, I imagine as most people have already stated, she will be know as Lili, which is pretty. Diana was an obvious choice for a second name. Doria, which is also a pretty name would have slipped in well after Diana but maybe they didn't want to go for overload.
I'm sure she's a lovely baby, wishing her a long and happy life.