My car radio resets itself, losing all saved stations and clock settings. This happens randomly and frequently during very cold winter nights. What could be causing this?

Not knowing what kind of car or what kind of radio, I could be off, but I think the most likely problem is a broken or loose power connection. Many radios lose their presets if power is interrupted. This could be at the fuse box, or inside the radio or somewhere in between.

If you can get at the radio wiring harness and identify the continuous power input wire (the one that feeds the clock and presets is different from the switched line that powers the radio), you could try running a new wire from a different unswitched fuse or a new fused line directly from the battery into that line, perhaps using one of those clip-on wire tap things. If the problem goes away, you know the problem is not the radio, but somewhere between the battery and the radio. A loose connection inside the radio is harder to diagnose.

My aftermarket Kenwood car radio reset itself as well. It seemed to do this during colder mornings, when the car "hesitated" to start. This really started to piss me off since I lost all my setting ( including radio, Sirius, color displays…etc.)
Well I think I just fixed this by adding a 4.7uF electrolytic capacitor on the ignition wire (yellow) to the radio.
I’ll explain for all those non-technical people (very simple). Basically my theory is that power was being interrupted to the radio and the Yellow wire was not holding the 12V. Adding the cap across the 12V and gnd (black wire) adds a small amount to stored power if that voltage drops. As of this morning it was a chilly 29 degrees and it seemed to work like a charm. Normally the radio would reset itself during the initial crank of the key

I had the exact same problem let me tell you how to fix it take your radio out an turn the switch off and get a test light find a wire that comes on without the switch being on when you find one hook the radio power wire to that wire but from now on when u cut the car of you will have to turn the radio off with the power button but it works