Mojang could fix this issue by creating a new authentication service.
It wouldn't necessarily just be for login authentication, it could be for any type of problem that requires player-specific authorisation.
From Mojang's POV it would be used like this:
Someone requests authentication.
Mojang sends a random code that needs to be used (to identify this request).
A user logs in to "minecraft.net/auth" and then specifies the random code.
Mojang then sends the UUID of the player that authenticated itself to the original requestor.
Here is an example on how it could be used by alpha and beta minecraft server owners:
A player joins the server.
The server requests that the user needs to login to "minecraft.net/auth" and sends a random code that needs to be used.
The player then logs in and specifies the random code.
Then the server gets a response with information about the player that has logged in.
I think this would be a great addition to the Mojang API that isn't necessarily just for old minecraft servers.
Samuel Fanson and Himblez your issue is not the same as this.
As Alex said this issue is for https://login.minecraft.net/session?name= and http://www.minecraft.net/game/joinserver.jsp?user= which has been shut down for a long time.
Mojang took those out of service for their newer authentication server http://session.minecraft.net/game/joinserver.jsp?user= meaning those old versions of the game don't have authentication at all.
It's quite clear that they aren't going to bring the old authentication servers back online.
My solution would actually solve this issue (and more), the bukkit plugin AuthMe doesn't help with anything except protecting other players from joining with your name. This solution would help preventing players from using alt accounts, making it much harder for them. Instead of them just changing their name they would need to pay to buy another account. And it would also solve the issue of players changing their names, since we could fully switch to using UUID's instead of playernames making name switching seamless.
So in summary it fixes:
Authentication
Alt-accounts
Name changing
And probably more
The solution I proposed requires very little effort from Mojang, and solves several issues from our end.
I think this is exactly what we want.