Q:

What primitive electronics are possible using boiled potato power?

A boiled potato battery provides power at $9/kWh, at 1/6th the cost of kerosene, and 1/9th the cost of a D battery. One single boiled potato provides 5 volts of electricity.
Pre-Columbian Americans had access to both potato technology and boiling technology, as well as primitive metallurgy (especially the goldsmithing in Central and  South America). Imagine that, once their leaders have seen the life-changing effects of electric technology, even the most conservative among them would sanction research into spud tech.
With the only change to their technology being the discovery of electricity via potato, how could life in a pre-Colubian Native American society be transformed?

A:

Very little, if any.
Such battery requires thin wires. The metallurgy of pre-Columbian America was significantly underdeveloped even by contemporary standards. Without a huge leap forward, they wouldn't even be able to produce one. 
Even without this constrain, their technological level at the time was insufficient for any use of electricity other then the most sporadic ones (like lighting fire). Even Europe/East Asia (which were centuries ahead from a technological perspective) wouldn't probably affected much. 
In reality electricity itself isn't as interesting as the things it powers. 

