To do this, you need a needle with a wide eyelet. You can find these at Jo-ann's, Michael's, or other craft/fabric stores. You may have to ask for an embroidery or curtain needle.
Once you have the needle, you pull your string through the needle, and then put the needle through the stuffed animal. You'll be making two holes in the stuffed animal; one where the needle goes in and one where the needle comes out. You may have to heat the needle over a lighter if it wont go through the stuffed animal easily. If you can't get a grip of the needle, use an anti-grip mat like people use to open jars. Then once you have the string coming out of the two holes, you can weave those strings through the cuff to secure it, and tie them wherever you can. it will depend on the cuff/where the holes are in the animal.