You can use 1/8 inch elastic cord, but I prefer drawstrings that do not stretch myself, although I usually sew elastic into the swimsuits that I make.

You could cut a length of T-Shirt like knit fabric and sew it into a long tube making sure the knit goes in the right directing for stretchiness. The ends could be hand sewn shut or simply tie the ends in a knot. Or you could attach two tubes of a matching fabric sew them to either end of 12' or 18" of ¼' flat elastic. The elastic section would be centered in the back of the drawstring pocket and not show on the front.