If you lived on the far side of the Moon, how could you infer the existence of Earth? Suppose that you deposit an astronomer, armed with our current knowledge of orbital mechanics, on a dome on the far side of the Moon, so that the Earth is perpetually hidden from them. 

(And, of course, assume that this person has no specific knowledge about the system they're in beyond what they can glean from observations. If you will, imagine that they learned all our modern orbital mechanics and related physics in alpha centauri, and then got teleported to our Moon.)

Now, it is reasonable to expect that this person should be able to deduce from observations of the sky that the body they are on is one half of a binary system, and they should be able to measure the orbital characteristics (semi-major axis, ellipticity, inclination) as well as the  position of the barycentre (much closer to the other body, corresponding to a much more massive partner). What observations are needed to deduce this? What level of observational accuracy is needed for those observations, and to what historical epoch does it correspond to? (I.e. would Tycho Brahe's kit have been sufficient? Would Galileo's? Would the ancient Greeks'? Or would this require a late-19th-century (or even later) observatory?)



(As pointed out in MartinV's answer, our astronomer might find it hard to distinguish between situations with an orbiting pair vs one single huge body. Thus, if convenient, you can assume that, via short ~100km forays from the dome, our astronomer is able to measure the lunar radius by measuring solar inclinations at different points with known distances between them, à la Erathostenes.)
A body tide seismometer on the lunar far side would pick up both the solar tide, and the 20 inch body distortion produced by the earth.
While "tidally locked", the moon is not in a perfectly circular orbit, and also wobbles a bit; libration. Your seismometer should pick up both effects.

Watching the parallax of mars cycle every 28 days, as suggested in comments above, might be a simpler way to go.