The best way to determine cast on stitches is to measure the circumference of your niece's foot around the widest point (this site
has a picture of where to measure), then make a gauge swatch with your yarn and needles. Measure the number of stitches in an inch, and multiply that by the circumference. Since most people like their socks fairly snug, you then multiply that number by, say .9 (a number between .85 and 1 usually works), which will make them slightly smaller than the circumference, so they'll be a little snug. If you're doing any kind of ribbing (1x1, 2x2) make sure the number is divisible by 2 or 4.
Phew! Hope that helps!