Is it a way I could change My Avatar's name?

[MrsLaNaiJohnson]

I want to know if I could change My Avatar's Name?

[Codewarrior Congrejo]

you can change your 'display' name:

Setting or changing your display name

You can change your display name once every seven days, but you can Reset it at any time, which makes it appear as your "classic" Second Life first name and last name, or your username.

To set or change your display name:

1. Log into Second Life.
2. In the Second Life Viewer, click Me > Profile... or  in the toolbar.
3. Click Edit Profile.
4. Click Display Name.
5. Type the desired display name in the field under New Display Name.
6. Type the name again to confirm.
7. Click Save.
   
   
   

For more infos look here: http://community.secondlife.com/t5/English-Knowledge-Base/Usernames-and-display-names/ta-p/700173

But be informed that the display name is not the same as your username. The username will allways remain the same.

[Rolig Loon]

Nope.  Your username is like pattern baldness or your Social Security Number.  It's yours for life.  You can decide to use a Display Name for unofficial things, sort of like using a nickname among friends in RL, but the few places that require your username --- like login -- will accept no substitutes. Your unsername is a unique, unchangeable identifier. See http://community.secondlife.com/t5/English-Knowledge-Base/Usernames-and-display-names/ta-p/700173