though they may not have the huge cranking amps that lead-acid batteries have, if you're willing to wait a few minutes, the Lion battery should be able to charge the vehicle's lead-acid battery to the point that you can start the vehicle
You do want to charge the lead-acid battery to more than the bare minimum though, because discharging a lead-acid battery too deeply permanently damages it. So some kind of voltmeter is a good idea.
Gasoline
This isn't a battery type, but it is a valid source of electricity in an emergency situation.
They're really REALLY expensive. Also, I live in a dense city and the noise level could be an issue with the neighbors.
vehicle alternator — You already have something that looks like an electrical generator in your garage, you just have to buy an inverter for it.
You're limited by how many amps your alternator can produce at idle, because going above that will drain your battery, so it's not worth getting an inverter with a wattage much above that, unless you're planning to regularly use the inverter while driving.
You might be able to change your idle RPM in emergency situations, but some sources say that can't be done on modern vehicles (without making serious tweaks to your ECU anyway). (TODO — what's the real answer here?)
Of course, if you're able to use 12 volts directly (for example, many portable power stations can be charged from a 12 volt source), then a separate inverter would be unnecessary.