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Try booting the mini without the display connected, and then, once the system is running, connect the display. This will often force the system to default to a low resolution, and then correctly detect the display when it is connected.
If that doesn't help, with the display connected, boot into safe mode (hold the shift key down when starting up) and once the system is running, change the resolution to 1024x768, and ensure the correct refresh rate is selected. Rebooting normally ought then to give you a good display.
If all else fails, boot into safe mode with the display connected, and then in the Finder's file menu, use the 'find' option to search for 'windowserver' (no quotes). You should get 2 or 3 results, including one with a string of alphanumeric characters. Move all of them to the trash and restart normally.