Thanks to Cathy for this Frugal Baby Tip!
Your best bet would be girl's underwear elastic. It is very soft. This is how I did it:
I measure the distance I plan to put the elastic, cut one piece of elastic this length and then cut the elastic piece in two. Fold one piece in half and pin the centre mark at the crotch of the shaped diaper - along the edge of the diaper. Then take each hanging elastic end, one end each way, stretch it along the edge of the diaper and pin the elastic ends in place, folding under a bit.
Place diaper in your sewing machine, stitching back and forth a time or two at one end of the elastic. Then, with your needle in the fabric, stretch out the elastic again and slowly stitch along the edge to hold the elastic in place. Stitch back and forth a time or two when you reach the end. Repeat for the other side of your diaper.
Hope that makes sense...