Go buy some very smooth, almost-satin fabric. Something polyester or rayon, and hoefully the same color as your strap. Cut the fabric longer than your strap, and 2" wider.
Get some shoe glue, and glue one edge of the fabric 1" into the strap.
Then fold it over, and spread glue 1" wide onto the other side of the strap. Then fold the last edge over to finish it up, being careful to fold the fabric just right so the second edge creases and meets the edge of the leather strap straight and flush.
I know this image doesn't show it, just cuz I'm trying to show a cutaway of what the fabric and strap should look like. But make sure that your fabric is far longer than your strap, so after the glue dries you can trim the excess at the ends.
And if you know someone with sewing skills, have them run a stitch around the entire edge to reinforce it. If you don't know anyone, take it to a tailor. They should be able to sew through leather. If not, you know they're not a good tailor!