I am assuming this is a tumbled stone, not cut as a necklace stone with a flat back and definite shape. If so, let me know and we will go one more time.
Here is the answer for a tumbled freeform stone.
The kind of Name Necklace part (called “findings” in the trade) you want is a bell cap. There are different styles of this. You should be able to find these at crafts stores and if you want gold or perhaps sterling, a necklace should be able to order for you. A few crafts stores will have sterling findings but most will have only plated items.
To attach the stone to a bell cap, a good adhesive should be used. There are two primary types of adhesive for this purpose. The first is 2 part epoxy cement and the second is a thick single part craft glue. There is a difference in the two: Epoxy takes more set-up time, meaning the cap must be set-up in place on the stone while the epoxy cures.The cement has no strength until some time has passed after the two parts are mixed. The epoxy has the advantage of curing with an almost unseen line and Custom Name Necklace. any kinds of cementing jobs.