Could a planets magnetic field magnetise the core of its own moon? Could a planet with a magnetic field make the core of its moon (or another celestial body), if it was made of iron say, into a giant permanent magnet? 
 A: I think that's a really good question!
"Soft" magnetic materials on a moon could be magnetized by a planet's magnetic field, but mostly temporarily.  On the other hand, if the moon's metallic core were molten and gradually cooled in the presence of a mostly-unchanging magnetic field from a planet, it could become permanently magnetized.  However, what complicates this scenario is that it takes a long time for a molten core to cool down, while the magnetic field of a planet (at least of the Earth) changes frequently because it is not a permanent field; it is generated dynamically by convection within the core and mantle of the Earth.  That is a chaotic process, so the field actually reverses every once in a while.  That means the field "frozen" into the moon's core wouldn't "know" how it should be oriented.  You might be interested in reading about paleomagnetism
I think yours is an interesting question because it might be possible to narrow down the dates of lava flows and the like on the Moon, based on the magnetic orientation of the solidified lavas and on the known geomagnetic history of the Earth.
