Specifically a radio telescope on the far side would be advantageous WRT low radio noise.
Some kind of interferometer with one antenna on the earth and one on the moon would be interesting. Yeah more convenient to launch two satellites into opposite sides of the earth's orbit or something, but this is close enough to have human repair / maint / installation techs at both sides.
One nice feature of a base on the moon is single stage to orbit is technologically challenging on the earth but on the moon its no big deal. Ditto "no biggie" to build a rail launcher for non-living cargos or an elevator for cheap living cargo. In the very long run, there will be people/things in orbit occasionally eating luxury moon produce because it'll be much cheaper to orbit from the surface of the moon than the earth, or the moon will be the breadbasket of space, sorta.
You get 24x7 coverage of half the surface of the earth... I suspect scientists and soldiers will have lots of telescopes pointed at the earth.
Unlike robots or probes if you have humans with enough duct tape onsite, human lunar jobs can be changed in minutes instead of decades for dollars instead of billions of dollars implementing a new mission from scratch. Although never underestimate the ability of mismanagement to screw things up. Just like on the earth surface, doing the same boring predictable thing 24x7 for years unchanged is best done by a robot, and "hold my beer and watch this" or "hmm that result is odd" is best done by a human onsite.