AVR's ADC modules are quite unique among 8-bit mcus (and even among 32-bit mcus) in that it not only offers differential capabilities but has a variable gain module there, allowing it to achieve better resolution than spec'd.
The Attiny's ADC module, unfortunately, is a little bit crippled in that its positive input must be at a higher potential than its negative input. Other AVRs, like usb128x or 32u4, don't have that limitation.
Unfortunately, usb128x is quite expensive. 32u4 is used in many of those cheap Pro mini boards. Unfortunately, those designers did not route out the ADC0/ADC1 pins.
The best pick for this would be a Leonardo board - its ADC0/1 pins are full accessible. But it seems an overkill for this purpose.
If you want to build, however, a milliohm / millivolt meter with digital display, a Leonardo + lcd shield would be a perfect choice.