Why rechargeable batteries use 1.2V
Not all rechargeable batteries have a nominal voltage of 1.2V; it is specific to a few chemistries that happen to be popular. According to Wikipedia, the following rechargeable battery chemistries have cell voltages of 1.2V:


Nickel-iron
Nickel-cadmium
Nickel-metal hydride


At a glance, it would appear that nickel is the common denominator, but this is not the case, as nickel-hydrogen and nickel-zinc have voltages of 1.5V and 1.7V, respectively.

So, excerpting the relevant sections of Wikipedia:

Nickel-iron:


  The open-circuit voltage is 1.4 volts, dropping to 1.2 volts during discharge.


Nickel-cadmium:


  Ni–Cd cells have a nominal cell potential of 1.2 volts (V). ...a Ni–Cd cell's terminal voltage only changes a little as it discharges.


Nickel-metal hydride:


  A fully charged cell supplies an average 1.25 V/cell during discharge, declining to about 1.0–1.1 V/cell...the starting voltage of a freshly charged AA NiMH cell in good condition is about 1.4 volts.


In conclusion, the 1.2V is only nominal voltage. 1.2V was a round number close enough to the typical voltage of those three chemistries and thus became the voltage on the label. However, every battery chemistry has different characteristics including voltage across a discharge cycle and open-circuit voltage.