Currently there is no way to hack the Wii Mini since all existing softmod options require an SD card. Maybe in the future it will be possible, I remember reading a thread on another forum saying it might be possible to solder an SD connector to the motherboard, but that's all I know.
Dr. House @ Wiihacks
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Yep, clearly states in our softmod guide the Mini-Wii cannot be hacked due to lacking traditional Wii hardware. Thread closed and will move into regular Wii newbie section (Mini Wii <> Wii-U, lol).