For little girls (and boys for that matter) the term will mean whatever YOU define it to mean. Judgements and prejudices towards the word are not born in your children - they come from how you parent and use the words. So as long as it doesn't mean spoilt and pampered to you - it won't to them. I told DSD a few months ago that when I was watching her at swimming I thought "wow she's so besutiful" DSD said "aww" and I clarifed by adding "and when I say that I mean strong and confident and lovely and nice and just a gorgeous person" she said "I know". It was a really lovely moment - unintentionally showing her that beauty is more than just the way she looks.

Princess will be the same - as long as it means something negative to you - it will to them - so.... Just make it mean what you want it to mean - a lovely person who cares about other people and looks out for people in everyday life - or something.