Hi Charles,
Firstly may I encourage you to post directly into the PICaxe forum. Of course there's no guarantee how long it will remain, but there are numerous posts on this forum going back for more than 15 years which make links to external web pages that no longer exist (and not just on ebay).
Yes, there are several ways that the resolution of PICaxe's ADCs can be increased. The simplest is to use FVR_1024 as the ADC reference to give 1 mV resolution. It's deprecated in the Microchip data sheet (presumably because they won't guarantee monotonicity) but it seems a useful trick. Secondly, I was surprised to discover that as the PICaxe supply rail is increased from say 3 volts up to over 5 volts, the internal noise appears to increase far more than proportionally. So at 5 volts there is sufficient internal noise on the ADC measurements that simply accumulating multiple measurements can give a genuine increase in the resolution.
However, a trick that I've often used, employs the on-chip DAC (which is basically just a 32-tap resistor divider chain), not as a precise divider but more as a "randomiser" or noise generator. It can increase the resolution (NOT accuracy) of measuring the PICaxe's own suply rail (implemented natively with the CALIBADC10 command to around 20 mV resolution) to better than about 1 mV. An example is given in the screenshots at the foot of post #24 in
THIS THREAD, in which the supply rail ramps down very slowly whilst monitoring the voltage using various methods, from the native READADC10 (CALIBADC10), through accumulating multiple measurements (with the "available" noise), to the DAC randomising technique.
Cheers, Alan.