Basic question related to relay operation
Can we make a slight change in the principle of operation of relay by
using a low resistance non ferromagnetic conductor as armature of
relay and due to low resistance a high current induced (Eddy current)
in the armature when we turn on or off the supply and a transient high
repulsive force(lenz law) acts on the armature and hence we can open
or close the switch ?

You can make things move (and slow down) by using eddy current induction but, in your example, this only works for the small period of time that the changing magnetic field in a relay coil lasts. Maybe a few tens of milliseconds.

Or there is any practical problem in this operation ?

Once a stable magnetic field is set-up, eddy currents disappear and the thing that moved (if held in neutral with a spring as per a relay) then returns back to its neutral position.