The Mini was waaaaay before Gadgeteer. So it’s not for that - it is however for a connector that GHI called UEXT, but that doesn’t matter… you can’t really find anything that’s meant to connect to it anymore
If you want to connect a specific module like the music module on a “generic” board, you need to delve into what the required pins are for that device. The product info sheet says this is an S module, and the wiki GHI Electronics – Where Hardware Meets Software shows an S module uses up to 7 pins (there are the three GPIO pins, including one interrupt capable one, plus the SPI specific pins CS, MOSI, MISO, and SCK). The product page also shows that it doesn’t consume power on the 5v line so you only need 3v3 and GND connections to power the module. So what you could do here is connect the music module to a breakout http://www.ghielectronics.com/catalog/product/405 that means you can easily access the pins (the Gadgeteer spec is .05" spacing not the usual .1" spacing on the mini), then wire the pins to the appropriate pins on your mini. Then you’d have to take the module driver code and try to compile it to NetMF 4.1 on a USBizi/Mini since you can’t use the full Gadgeteer core on that platform. Certainly not impossible but not easy like if you had a Gadgeteer mainboard.
As for the Analog pins, some e-blocks can be used (but I don’t have a mini to know how the pins are exposed). But if you look in the forums there’s a few issues with 5v power destroying pins so personally I’d just try to buy a LDR and connect it in a voltage divider setup to an analog in pin.
(Ah I note @ Architect has helped out too - note you’re just going to have to wire things yourself to get them to work on a Mini)