Quoted from Wikipedia, Lenz's Law states that "An induced electromotive force (emf) always gives rise to a current whose magnetic field opposes the original change in magnetic flux." In slightly simpler terms: nature abhors change and fights to oppose any change in magnetic flux by creating its own counter-current in the direction opposite the change (not that it is opposite the change, which may or may not be opposite the flux or magnetic field).
If a magnet is moved inside the solenoid, the flux changes, which induces a current in the solenoid.
If the magnet comes to rest inside the solenoid, then there is no change in flux, even though there clearly is both a magnetic flux and a magnetic field. Since there is no change in magnetic flux, there is no induced current.