Free email
The vast majority of people use free email boxes on Yandex.Mail, Rambler-Mail, Mail.ru, Gmail, etc. On these services to change the username for the registered account is impossible. Accordingly, it is impossible to change the email address. Therefore, in the case of the free mailbox can only register a new account with a new email address.
If you want to continue to receive mailings that come on the old mailbox, you can forward emails. For example, on Yandex.Mail to do this, go to "Rules for processing incoming mail and create a rule forwarding of letters, putting the address of your new mailbox.
Paid options
Paid options there are two. You can buy your own domain and set up a mail server. In this case you will have virtually unlimited freedom in choosing a new email address. You can select and domain name, and a prefix in the email address. For example, ivanov@ivanov.ru, neivanov@ivanov.ru etc. Disadvantage of this option is that you will need to solve all the problems related to domain and hosting or to hire a third-party specialist. But, if you ever again need to change the old email address to a new, you do so already with ease.
The most convenient is another option. It implies using the services of a paid email service. One of the largest international services is FastMail. The cost of servicing the account on FastMail starts from $ 10 per year. For that money you get a Inbox with a powerful system of anti-spam, smart filters to handle mail and 250 MB of disk space for storage of emails.
FastMail allows you to change once the username (that is, a prefix that appears before the icon “@” in the email address). If you decide to change your address, you will have to contact customer support.
But not 10, but 40 dollars a year FastMail allows you to use as your email address own domain. Just go into the control panel of the domain and specify there the DNS server FastMail. Then will remain in the FastMail settings to specify a domain and a new email address (for example, petrov.ru and mail@petrov.ru).