If you want a simple "flat" strap, just knit a long strip. CO however many stitches wide you want it and then just do a plain garter stitch (knit all stitches) until it is as long as you want. Other alternative would be to do an i-cord. I JUST completed my first one of these for a project and it is super easy. There is a video on this site that shows you how.
Best of luck and happy holidays!
OTN: my first attempt at any type of mitten - doing fingerless
Oh...and I'm heajames on Ravelry