Depending on your budget and expectations there are two ways to approach naming a new rose:
Naming a rose that will become widely available:
It may be possible to get a commercial rose named in memory of a loved one by entering into an agreement with a breeder and paying them a fee. However, rose breeding is highly skilled and time-consuming; and the cost is likely to be in excess of £2000, and for very good ones more than £10000.
For further information see The British Association of Rose Breeders.
Naming a rose for private use:
Non-commercial roses for sharing with your friends and family can be obtained from several sources (see below). These roses will not be available in garden centres or nurseries and so unless propagated from cuttings there is always a risk that they will be lost to cultivation.
Although roses offered through this method may not have shown the promise to become commercial varieties, they will still be unique and can be attractive nonetheless. You might like to check that the name you have chosen will be accepted by the American Rose Society which maintains a list of all valid rose names. Expect the cost of such a rose to be between £50 and several hundred pounds.
There are a number of companies and organisations which advertise this service:
British Association of Rose Breeders
World of Roses
Country Garden Roses
Please note: The mention of any supplier or service does not constitute an endorsement by the Royal Horticultural Society.