I believe children in Waldorf schools routinely make knitting needles before they learn to knit. It brings a little bit of woodworking into a fiber-based activity, and it is calming to sand and finish. Calm is a good thing to have when knitting!
There is also a sense of accomplishment in having created one's own tools before beginning. When you've put time into polishing your tools, you are more invested in using them with care.
A 1/4" dowel will make size 10-1/2 needles. A 3/8" dowel gives you size 11 needles. Both of these are for use with bulky yarn and are easy to learn to knit with.
Mark and cut the dowel to a size you like--I think 12" works well, but I've seen children's needles that were shorter, and if you are making something wide, you may want a longer size.
You'll end up with two pieces of dowel roughly the same length.
After this, you need to make points. You could whittle--
--but an easy way is to use a pencil sharpener. You don't need, or even want, your knitting needles to be pencil-point sharp, just tapered.
In theory, you could knit with pencils themselves, although you'd be getting lead all over your project. So let's stick with the dowels, which are now sharpened and ready to sand.
We start with 100 grit sandpaper and gradually move up to 320, sanding until the wood is very smooth.
When you are finished sanding, you simply need an end cap of some type so that the yarn won't fall of the end of the needles. Pink pencil cap erasers work just fine, although you'll want to glue them so they stay on. K and E glued acorn caps to theirs:
S rubbed his needles with beeswax and orange oil, then sculpted balls from fimo clay, baked them, and glued them on:
These were dyed with colored stains from Ikea, coated with tung oil, and capped with Lego pieces that had been drilled out:
I'd love to see other examples of homemade knitting needles, and will gladly add any photos you send to this post.