Edit or change iCloud email using current name

Do you mean the @icloud.com email address rather than the Apple ID/login address?

Once you have created an iCloud account and chosen an @icloud.com address to go with it you can't change the address (short of creating an entirely new account).

However, all is not lost. You can add up to three 'email aliases' - these are additional addresses (not accounts) which deliver into the same inbox as the main account. (In fact it's a good idea to give out alias addresses, rather than the main address, because if they attract spam you can easily change them.)

You should be aware before you start that once you've created an alias you cannot turn that address into a full iCloud account or move it to another account.

More information on aliases here: https://help.apple.com/icloud/#mm6b1a490a

If you do in fact mean the Apple ID/iCloud login:

If your primary address (the ID) is an @icloud.com, @me.com or @mac.com address you cannot change it. If it is a non-Apple address you can change it as follows:

1. Firstly, sign out of iCloud on all your devices: go to each of your devices and sign out in System Preferences (or Settings)>iCloud - 'Sign out' on a Mac, 'Delete this account' on an iOS device (this will not delete the account from the server). If you have 'Find My iPhone' etc enabled you will need to disable it first. Your synced data will disappear from your devices but will reappear when you sign back in.
2. Create a new email address, for example at Yahoo or Gmail, or anywhere convenient (or you can use an existing address as long as it is not associated with an Apple ID). You can change it to another Apple address only if it's listed when you click the link to change (in practice this means your main @icloud.com address, and the equivalent @me.com and @mac.com versions if you have them: it does not include email aliases).
3. Go to http://appleid.apple.com and click 'Manage your Apple ID'. Sign in with the current ID.
4. Where it says 'Apple ID' and gives your current ID email address, click 'Change your Apple ID'.
5. Enter your new address and click 'Save changes'
6. Sign back into iCloud on all your devices with the new ID and password. Your synced data will reappear (give it time).

I reiterate - before you start, sign out of iCloud on all your devices. Failure to do this will lead you into difficulties.

This information may help you: Change your Apple ID - Apple Support