Loss of energy when permanent magnet falls trough coil?

Part of the energy will be used up in driving eddy currents (induced currents) in the coil, so the final energy of the magnet will not be $mgH$. The total energy of the magnet $+$ coil system will, however, be conserved. Exact calculations will require knowing the geometry of the coils, the material they are made out of, etc. and solving Maxwell's Equations.

i think a picture says a million words in this case.
mgH = 1/2 mv² + E_disspated.

obviously, you would need maxwell's third equation (faraday's law) to calculate the induced voltage on the coil. I assumed too that there is no air friction and that the energy consversion from one to the other is 100% efficient with no loss whatsoever.

Assuming no friction (from the coil or air) and no resistance in the wire, the difference between the mgH at the top of the coil and the mgH at the bottom would indeed be equal to the amount of energy transferred via eddy currents into the coil. However, unless done in a vacuum, there would be energy lost to air resistance, and no conductor has no resistance, so it would be very hard to work out exactly how much was conserved. That said, an approximate answer for how much of the mechanical energy was lost may well be obtainable from Maxwell's equations, and potentially it would be possible to find out how much had been lost due to friction and resistance as a missing figure.

Hope this helps