How do I restart probes that have lost power?

You can transfer power from batteries just like you can with fuel. So a rescue ship with a Claw can connect to the probe. Load it with batteries, transfer enough juice to get the panels extended.

A nice efficient engine, and a go-anywhere delta-v - a handy thing to have in orbit when things go awry.

Short of launching a (possibly manned) rescue mission and connecting the probe ship to a craft that has power, no.

Mind, to even have a chance at this the original probe needs to have some kind of docking-capable ring on it in order to connect properly, but if it does and you manage to dock a rescue ship to it, both craft will be treated as one in terms of electrical power and the probe core should boot back up. Don't forget to turn the solar panels back on before detaching again, though...

It depends on how the probe was designed. If it doesn't have any solar panels, docking ports, or a cockpit + some fuel, then there's no restoring power to it in the vanilla version of the game.

That being said, when designing a new probe:

• Always add solar panels, preferably on 4-6 sides. It's possible to lose power to a probe equipped with solar panels and not get any power from them because of the orientation of the probe when it lost power.

• Add a docking port. A separate ship/probe can dock to a dead probe and transfer power to it. Additionally, having a docking port on the probe means you can add to the probe while in space, potentially adding more batteries or solar panels.

• Since engines also restore battery power while running, a dead probe could technically be recharged by having a kerbal climb inside a cockpit and engage the engine.

If you really want to save the probe, you can edit your save file and fill it up with electric charge. I haven't played KSP for a while so my memory is foggy and I don't have it installed on this machine to verify the save file structure, but basically:

1. Save your game, either quicksave or a named save.
2. Find the save file in /Saves directory (back it up in case you mess something up).
3. Find the VESSEL by searching for its name (might be easier if you rename the probe to something unique first to avoid " Debris" vessels).

4. Within the VESSEL definition, find a the empty electricity resource

   RESOURCE
   {
       name = ElectricCharge
       amount = 0
       maxAmount = <some number>
   }
   

5. Set the amount to whatever you want, save the file, and load it in KSP.