Disclaimer: This is my first one, so if you have feedback about what I could post differently next time, I won't be offended, but please be gentle (and I'll thank you now for making my next Instructable more instructable :)
Step 1: Stuff you'll need
Those left over dowel pieces/remnants
A pencil sharpener with a hole big enough to put the dowel in (or an Exacto Knife if your dowel is too big for the sharpener)
Several grades of Sand Paper (Coarse to fine)
Some kind of oil (lavender/olive)
Step 2: Sharpen the ends
If you want double-pointed needles, follow this step for the other end of your dowel.
Step 3: Sand the dowel
Start with the coarsest sand paper you have working your way to the finest until all the splinters are gone (you don't want that precious yarn fraying before the project is even finished! nor do you want the knitter's hands bleeding all over their expensive cashmere!)
Step 4: Embellish and Treat, as needed
When you're done decorating (or if you chose not to), rub any exposed wooden surfaces with oil - it will take care of any hanging shaving fuzz and give it a nice shine, making your new knitting needles look nice and pretty. (Tip: To prevent from having an overly greasy needle, instead of pouring the oil directly on, I put a few drops of the oil on my palm and rubbed the needle with that greased hand.) You might have to let it cure a few hours and do this process again to get a well-conditioned needle.