i don't know about child friendly search engines, but several good sites have sections for kids, like national geographic for kids, BBC kids, CBC kids, the government of Canada etc. or are you looking specifically for a database for kids?
it is such a hard time when they are in that transition! unfortunately i think it is really child dependent how long it takes them to get used to not napping, but i can't see it lasting a year :)
hope it gets better soon!