Active highpass filter for microphone, how to choose cutoff frequency?
I was thinking 60Hz would be induced from AC wires around the house
etc. So triple that at 180Hz should be good - I guess. I would
appriciate any input and help on the matter.

Well induction from AC wires is due to current flow in those AC wires inducing voltages in series with the microphone wires but, the current in the AC wires can easily have multiple harmonics of 60 Hz and you'll find that the third harmonic (180 Hz) is still very dominant in household appliances etc..
Fifth harmonic is 300 Hz so, where do you conceivably stop?
Answer: you use good practises and avoid induction by twisting signal cables so that the net magnetic induction is basically very low. We also use shielded (or screened as they say in the UK) to remove/reduce electric field coupling. In other words, we filter as a last resort but if we are at that last-chance saloon, then surely we have made a wrong design decision a few stages ago.
Sometimes when I mix guitar or vocal tracks from one of my pals in the US, I use a notch filter to remove as much 60 Hz as I can. Reason: his setup is carelessly put together and I don't think his mancave/shed has a decent earth. Mind you, it's usually some trashy rock song we are writing so it doesn't really matter.