You can buy the "silky trim", it is called blanket binding and is sold in packages at sewing stores and comes in many colors.
Do a search on blanket binding and quilt bindings and you should find easy instructions for doing it.
Kathy G in MI

Did a search for you. Here is the link to instructions from Wright's, who makes bindings.
You may find that your son will not want the replacement blanket. My first child still has her "blankie", it is in rags. Very embarassing to look at. I would replace one side of the offensive blanket and one half of the trim on it. Once she got used to the "new" parts she would let me replace the next 1/2 of blankie.
BTW, she is now all growed up with 2 kids of her own AND she is a pediatrician! AND her blankie is now in a drawer. Hmmm, will have to ask if she plans on getting rid of it.
Kathy G in MI

Here is a link that might be useful: Blanket binding class