By basic do you mean stocking hat, skull cap, just one that pulls over close to the head, bonnet style?
I made this child's hat that stays on well when it's knit with DK weight yarn and smaller needles.
The pattern is the Tychus from knitty The pattern calls for size 9 US needles for an adult male hat. I used size 3 or 4 needles (sorry I can't remember which) and some DK weight yarn for a hat for a 1 year old. Just measure the child's head and compare the gauge. Do a strip of garter stitches to get your gauge (the hat is done in short row garter stitches sidesways). It has a LOT of stretch so your gauge doesn't have to be dead on.