Counterfactuals are complicated, but I didn't realize how complicated until I wrote all this. The basics have already been invented for other reasons (internal combustion engine, battery, electric motor, etc) so any sort of car should work fine aside from the infrastructure (power distribution, roads). Which sort is best depends on e.g. whether there's cheap oil (there might be if we hadn't used so much oil already) and what people are using cars for (people would be used to another mode with different capabilities; I've read that humans regardless of transit mode tend to prefer about an hour commuting per day and the distance of an hour depends on the prevalent mode). Random thoughts follow:
If the automobile were invented in 2013, we wouldn't have range anxiety, because we'd be used to getting around on horses. (Without range anxiety, electric cars might be the cheapest.) Or we would have range anxiety if there were lots of functioning places that would exchange your horse for one that wasn't tired (and no car recharging/refueling stations, of course).
Or maybe trains would still be the way everyone gets around. The streetcar would be ubiquitous, and would be the only reason we'd have quality roads (with rails in them). The streetcars might run on electricity or natural gas, judging by modern bus and light rail. Bicycles might be popular, and suburbs would have developed around streetcar lines rather than along interstate highways (I'm US-centric).
Cars might not need charging/fueling stations, if they could piggyback on an extensive electrical streetcar network's power lines. Then they would probably be electrical and have battery for short off-network trips.
The probable lack of a good road network is going to be a problem. We'll have to design a car that's good off-road. Preferably we will avoid annoying bicyclists by wearing away their dirt roads (if bicycling is in style and horses aren't); but we'll fail, because just having an enclosure for a person (I assume an enclosure is necessary to count it as a "car") is pretty heavy, and we can't improve wheels/tires terribly much.
Which parts and energy sources would be cheap? David Archer's _Global Warming: Understanding the Forecast_ (2007 edition, p.101) tells me that about 2/3 of US oil use is in transportation. The world might not have as much oil production capacity if the car hadn't existed (thus higher prices; then again, a newly invented car wouldn't suddenly be used by everyone, so oil production could ramp up over time along with demand); or the world might still have lots of easily extractable oil (lower prices).
Who would buy/use this car: rich? middle class? People who already travel a lot? People who are content? Individuals? Businesses? Transit companies (along the lines of ZipCar or self-driving cars)? Taxi drivers (there would probably be taxis powered by horse, bicycle, or some sort of engine)? How would people learn to drive this car? Probably most people wouldn't be used to operating a dangerous piece of machinery, and there wouldn't be laws regulating it (or there would be strict industry-focused laws).
The lack of experienced drivers suggests self-driving cars are a good idea; but the lack of well-marked roads means our AI tech needs to be a bit further ahead and we might have to wait 5 years.
EDIT: crossposted this response to my blog, since why not http://idupree.dreamwidth.org/2568.html