What would happen if we magically grabbed Mercury and put it into orbit around Venus? Would it help?

Well, the gravitational effect of Mercury would gradually change Venus's rotational period. This has already happened in the Pluto/Charon system, so that the two always keep the same face to each other, and the same thing is happening with the Earth and the moon. But would Venus speed up or slow down. It would only slow down if Mercury's orbit took longer than Venus's day (243 earth days). That's too far out for Venus to hold on to Mercury against the Sun's gravity. I did some quick calculations and I think the biggest stable orbit would be about 50 days long. So Mercury would slowly speed up Venus's rotation, which helps. (On the other hand, Earth's day is shorter than the Moon's orbit. So our day is getting longer and our own dynamo should be very slowly weakening.)

If Mercury's orbit was eccentric (and it would be, because the Sun would be constantly causing little disturbances) you'd also get tidal forces within Venus. The core would be stretched and pulled, generating heat. The same thing is happening in the large satellites of Jupiter- particularly Io where tidal heating creates tremendous heat that powers its volcanoes. More heat means the core will be more molten, and stay molten for longer. That also helps.

The big sticking point is convection. You still need a temperature gradient in the core and, as far as we know, tectonics is the thing to do it by releasing heat through cracks in the crust. I don't see how the presence of a moon can start plate activity on a planet that doesn't have any. I don't know if tidal forces alone can set up convection currents in a planet's core, but it seems doubtful.

I suppose, after all that, a moon might help a little bit if you want a magnetic field but not very much.

Venus as far as we know has a molten iron core, but its day is 243 earth days long which is only enough to generate a weak magnetic field. Also, it has virtually no convection in its core. This is because there is no temperature gradient in its core. On Earth, plate tectonics allow some of the heat to escape so that parts of the core are always cooling and there's a temperature gradient. Venus has no plate tectonics. Slow rotation and no convection = bugger all magnetic field.

Mars has a day short enough to power a dynamo, but it also has no plate tectonics, and its core is more solid than Earth's or Venus'.

UV light can slowly split water into hydrogen and oxygen ions. Since these are charged particles the charged solar wind can pick them up and carry them away because Mars has no magnetic field to protect it from the solar wind. The same this is known to be happening on Venus; it's gradually losing what's left of its water. Earth is OK because we are shielded by our strong magnetic field- the hydrogen and oxygen stay in our atmosphere and presumably turn back into water.