I purchased a RTH9585WF smart color thermostat to replace my mercury controlled thermostat. How do I add the blue common wire at the boiler relay where the two wire system is connected?
You'll either need to use a separate 24VAC transformer, or replace the boiler relay with something that provides a C terminal
Sadly, your Argo AR821, like many older boiler relays, does not provide a C terminal at all. As a result, you'll need to either wire an external 24VAC transformer to Rc and C on the thermostat to provide it with power while connecting Rh and W to the relay terminals, or replace your boiler relay with a newer model that provides a C terminal.
Pop the circuit board out and solder a wire to the back side.
We can faintly see a copper trace from the terminal where your red wire lands going toward the relay. Another trace is visible going from the terminal with the white wire toward the transformer. The red and white wires are reversed: R should be the one that goes to the transformer; W is the one that goes to the relay. It makes no difference whatsoever with a two-wire thermostat, but when you want to add that third wire for C, it starts to matter.
Common is the at the other end of the transformer, where there will be a circuit board trace leading to the relay.