Can't change my avatar on any device.

my suggestion would be to log into account.live.com to see if it prompts you to complete any account information. if not try to use the avatar editor on xbox.com make any change to your avatar(hair/shirt/boots.....etc) and then save the changes. power cycle
the device you are using and sign back in and try again.

Follow these steps separately, if one doesn't work do the next.

1.Make a change to your avatar, save and try again.

Note Any avatar change will do because the change causes the service to sync your account for your profile.

Launch the Xbox Avatar Editor app.

Xbox Avatar Editor

Sign in and make any change to your avatar.

Select Save.

2.Follow this step and try again but first if your subscription has auto-renew turned on, disable it.

Press the Guide button on your controller, go to Settings, and select System Settings.

Select Storage.

Highlight but don’t select a storage device, and then press the Y button  on your controller for Device Options.

Note It doesn't matter which storage device you choose. This clears the cache for all storage devices.

On the Device Options screen, select Clear System Cache.

When prompted to confirm storage device maintenance, select Yes.

I have been experiencing this myself for a couple weeks now. I have tried changing my avatar via 360, Xbox One, my PC browser, and even my Windows Phone and I get the same message. Not that it's a huge issue or anything but it should be brought to attention.

Make a change to your avatar, save and try playing again.

Note Any avatar change will do because the change causes the service to sync your account for your profile.

Launch the Xbox Avatar Editor app.

Xbox Avatar Editor

Sign in and make any change to your avatar.

Select Save.

Trying to make a change to your avatar might solve the issue:

1. Launch the Xbox Avatar Editor app .Xbox Avatar Editor
   
2. Sign in and make any change to your avatar.
   
3. Select Save.
   
4. Try to sign in on your Xbox 360 console or attempt to play your Xbox 360 backward compatible title again.
   

Note: Any avatar change will do because the change causes the service to sync your account for your profile.

Clearing the cache might also work:

1. Clear the cache:
   
   1. Press the Guide button on your controller.
      
   2. Go to Settings and select System Settings.
      
   3. Select Storage or Memory.
      
   4. Highlight any storage device, and then press Y on your controller. (It doesn't matter which storage device you select; the cache will be cleared for all storage devices.)
      
   5. Select Clear System Cache.
      
   6. When prompted to confirm storage device maintenance, select Yes.
      
2. Restart your Xbox 360 console.
   
3. Try again to redownload your Xbox Live profile.