Can low signal voltage damage a relay ? Probably, over time anyway.
I suggest you find out why the dash lights were dim as stated in your first post and why you later found less than 12 volts on that start signal wire.
Good luck
I seriously doubt that too low a voltage would damage a relay. It would either work or not. It's just an electromagnet coil which activates a spst switch. The main thing that would damage a relay is too much current through the switch contacts. My lame Ford Explorer was real good for doing such in that it was never designed correctly!