Which worlds in the Solar System are worth branch offices?

That is a matter of opinion.

Obviously a footprint on Mercury makes sense for solar power and weather forecasting due to its proximity to the Sun, and harvesting helium-3.

By contrast, a footprint on Triton makes sense for cryogenics research, superconductor-based computing, and other industries that benefit from low temperatures.

Rock mining is best done in the Main Asteroid Belt and The Kuiper Belt.

Earth’s moon (Luna) and Mars have their advocates for settlement.