Premium Members Can Now Change Their Names

Go to your Dashboard, Account >> Change Name.

At the time of writing, the fee for Premium accounts is $39.99 per change.
Just to make it clear, this is the new ability to change your actual avatar name, not the display name.

On Monday, April 13th, Linden Lab announced the return of Last Names to SL. Also known as Name Changes, the feature re-introduces the capability for users to select a new name, as the blog post explains:

Back in the day, Second Life Residents were given the option at registration of selecting from a variety of pre-determined “last name” options. The use of shared “last names” helped build community among Residents who found instant kinship and bonding amidst these newfound virtual family ties shared with strangers of the same lineage. Similar to the commodity of dot-com domains, some “last names” held a special status in the community. Some were extremely rare and, in some cases, there were perceived attributes and reputations associated with certain last name offerings.

This is only available to Premium subscribers, who may change their first name or their last name or both their first and last name whenever they wish. The first names are free-form, however, last names are selected from a list (with the available names updated periodically).