It's possible, but it would only be a duplicate of the D-pad, and it would require some programmable chip along with some coding. The N64 joystick in the official controllers uses gray code to work. This can not just be wired up to any other kind of controller or circuit and work, it must be 'decoded' and converted into what ever setup the GBA D-pad uses, most likely common ground.
Now you could use a 3rd party type of N64 controller's stick, as they are most ofter the 10k Potentiometer style, but even it will need to be taken apart and modified to work, and again it would only work as the D-pad and then only if the GBA uses a common type of button layout.