There's many threads on this, but here's a brief summary. I've stayed at all three.

- HR is in an office district in Shinjuku. Not the best location, fairly quiet and a bit of a walk to the actual station but like you said a good value especially on P&C.

- GH is in Roppongi which is the most foreign friendly neighborhood and the best location. You will find the most English speaking people here. Lots going on in terms of bars, restaurants and city vibes.

- Andaz is a few minutes (5min cab) from GH and a much nicer property though slightly less ideal location, however still very well located and close to the Toranomon subway line. You will have to take a connection to get to the JR which is worth considering if you plan on using the trains a lot.

Most would opt for the Andaz, you might also consider splitting your time between GH and Andaz as they're close enough that checking in and out shouldn't be a huge hassle.