Ok, I don't have time to do this (yet?) but, it could be interesting to see if it were possible...
Here is the idea: for cheap high resolution ADC abilities on a xxduino, say you could produce a variable voltage off one of the digital, or analog pins, perhaps via a simplistic DAC - 4 bit would do, you could measure the voltage from the read pin then alter the voltage on VIN via the DAC to narrow the band to be measured, and get higher resolution. The issue I could see it that it uses extra pins and I don't know you would alter the lower threshold of the ADC range - eg, move 0v to say 2.5
I realise I could do that, but that would require user input.
Huh? What do you mean?
In any case, there is no precision voltage source on the board, so whatever DAC scheme you have in mind would already have some built-in uncertainty in voltage due to the uncertainty in the DAC supply voltage.
Besides, I think what you are trying to get is higher accuracy, not higher resolution. Resolution is fixed by the microcontroller hardware, though you can increase effective resolution through averaging (assuming uncorrelated noise).
If you want higher accuracy, you need an accurate reference. And there just isn't one anywhere on the board.