The only way to fit lots of stitches on a needle is to use a circular needle and knit back and forth. The cord of the circ holds the stitches, but the knitting is the same as working on two needles.
You can really do any pattern you like. If you want to go with a blanket in stockinette stitch, you'll need to work the edges in garter stitch so you don't end up with a tube.
Knit the first 5 rows or so, then always knit the first and last 5 stitches.
For variety, you could work a basket weave--that's not difficult.
has lots of patterns.
I think I'd go with a machine washable worsted weight so it doesn't take until your little girl graduates.