Can an EEG be re-sampled to fix a poor voltage resolution?
The bit depth basically indicates how accurately your analog-to-digital converter records and reproduces the signal (Fig. 1). A higher bit depth means that more subtle fluctuations in the waveform are more faithfully reproduced in the digitized signal (source: Presonus)

As the answer from Bryan alludes to, data cannot be generated de novo, unfortunately. Data can be processed, yes, but data acquired at a certain resolution cannot be enhanced. Of course, as Strongbad indicates, you can generate an indefinite number of steps by upsampling, that's not an issue, but that will simply leave you with an estimate of possible intervening values; the real values are simply not there and are lost. Look at Fig. 1. If we would upsample the data, the points between successive samples would be interpolated, seemingly increasing your voltage steps, but the resulting trace will not be any better than it was before (only bigger :-).

Note that the resolution in the time-domain (sample rate) will not help you in any way. The amplitude (y) step size in Fig. 1 will not change, even if you has used an infinitely high sample rate. 


Fig. 1. Illustration of bit depth. source: Presonus