I bought some necklaces for a friends pups and they are made with the clasp. When I got them they were so cute that it got me to thinking they would be awesome to make as gifts for friends. I made a few, and since the ones I bought had the clasp, that's how I made mine. But I know Rhonda has the ones that are made on the stretchy string, and they look easier to make, and just as nice. There is another member here (Paige's Mom) that has some necklaces for her pup, and I'm not sure but I think the lady that made them has a website, or maybe she sells on Etsy. I think they are made on the stretchy string as well. Plus I think it would be easier to get the sizes right with the stretch. The first 2 that I made, I made them 1" larger than the neck measurements, and they worked out fine.
After I got the ones I bought, I went to eBay and Etsy and did a search. That will give you some ideas of what's already out there, and maybe help you decide how you'd like to make them. The lady that made the ones I got does a few different styles. She uses the stretchy string, and the clasp.
Anyway, I think both styles are nice. But the stretchy ones would be easier and cheaper to make. Plus you don't have to worry as much about the size being wrong.
Best wishes!