Reply to Billy Bug and maddieconman:
Yes, it is possible to name a rose, and an internet search will highlight many companies selling gift packs that include 'unique' rose seeds that can be grown into plants that you can name. This is a fun and affordable way to name a rose.
Naming 'The One & Only' rose was different, as this rose was bred by Meilland – one of the top rose breeders in the world. Breeding roses takes time and money, as does registering the name, so these costs have to be recouped from of royalty on future sales of the rose.
If you have a great name for rose, or would like to raise money for a charity or good cause from the sale of a rose, then start by contacting individual rose growers. They'll know exactly how this can be done, and what costs would be involved.
Commercial rose growers need to cover their growing and production costs, too, so are more likely to be interested if you can guarantee buying a large number of rose bushes.
Roses make a great gift, so choosing a rose with a meaningful name like 'The One & Only' gives it wide appeal.