What can cause sudden drop of my battery in Nexus 4? I have new Nexus 4 phone and I found it dead several times after the night. Obviously it wasn't used during night, screen was turned off.

After I plugged it to the charger I had to wait a few minutes before I was able to turn it on. Battery monitor's graph has a sudden drop from 60-70% to 0, what can be a reason for this?

See screenshot here:
What you're probably seeing is the charge monitor greatly mis-estimating the amount of charge left in the battery. It thinks there is a lot of charge left, but in reality there isn't—the battery is almost depleted. When the battery actually runs out, reality takes over, and the phone goes off.

You'd normally only see this when using a battery that is long past its lifespan. You also should see this during the day—or at least, you should see a much shorter than expected battery life.

You could try resetting to factory defaults (wiping the phone), which might help if its just a software issue.

I'd guess its a hardware issue, and you need to RMA the phone.