Just installed a new remote for a ceiling fan. Lights and fan were on and working fine one second, but then lights went out, but fan was still working. Bulbs are fine. Re-checked all wiring. Fine. I can hear "click" when I try to turn it on/off, but lights are not coming on. What could be the issue?

So you replaced both the remote and the receiver in the canopy, right?

Yes.
Old remote got chewed up by our puppy, so we bought replacement - receiver for canopy with a new remote. Easy job. Everything worked fine for about an hour. Then lights went out. Checked every bulb, every wire. Not sure what's the issue.