How can I send an email to Administrator when Users change their name?
yes in Rules you can achieve it 

Step 1 : In Events Event add "After updating an existing user account" 

Step 2  : Conditions Elements "Execute custom PHP code "

return [account-unchanged:name] != [account:name] 

or add data comparison and add 

Data to compare Selected data: account-unchanged:name

Operator : equals

Data value : account:name

Step 3: In action send mail 

refer this for same kind of problem with role. but you have is name 

How do I create a Rule &quot;When role is removed, send email to the user&quot;?