There are several ways to do this and so it becomes a matter of personal preference. The Russian join
works well with cotton yarn and it's shown under Tips on the Free Videos tab at the top of the page.
If you're using two strands held together for the blanket, you need not join both new strands at the same time. You can overlap one old strand and one new strand for a stitch or two and then weave in the ends. Both techniques give a double thickness of yarn for a few sts but it's unnoticeable.