Analog inputs with Intel Edison mini breakout board?
Use an external ADC connected to one of the digital interfaces of the Edison, for example I2C. Sparkfun sells PCBs that are designed to attach to the Edison in a stack, and here is one for a 4 channel (single ended, 2 channel differential) Delta Sigma 12 bits ADC. https://www.sparkfun.com/products/13327 I haven't looked around, there might be cheaper alternatives.

Nothing prevents you from making you own "edison block" with other ADC chips / analog multiplexers. It's probably cheaper, but very often your time is more valuable than a 10 dollars savings.