Battery technologies like NiCD, NiMH and good old lead-acid respond rather well to trickle charging from a small solar panel. Rechargeable Li-ion batteries seem to need more current to charge effectively (but keep in mind I haven't played with the polymer type). Offhand, I'd say you're probably looking for a panel in the 10-15 watt range, which is pretty big for backpacking. I suggest you check with the manufacturer for minimum charging current, safe range for input voltage, and recommended options.

For a car-based recharging system, there's no need to buy a separate power source since you already have the vehicle battery.
- You can get a 75W modified sine wave inverter for about $10. Plug it into your cigarette lighter and use the transformer your battery pack came with.
- Or, you can get a regulated DC power supply which will hold the voltage at 12 VDC fairly exactly. I'm not sure if any of mine will supply 1 A though (I'll check in a few minutes and post the results).

If you have a voltage meter, check what the raw output (no load) of the wall charger is. That will give you some idea of how finnicky the battery pack is regarding input voltage.

EDIT: The Xantrex (no affiliation) VDC regulated power supplies I have would probably supply enough current. I picked up a couple of these for $5 each from a clearance store, including multiple tip styles.