Your friend is wrong.
It’s the black laptop that should have longer battery life.
[Before going into detail, I should explain that the difference will be so small as to be practically unmeasurable, so I’m not trying to make a real case (heh) here. To all intents and purposes there will be no difference, but since the question has been asked (in the OP) and answered (by fishbicycle), here’s a slight diversion.]
Modern laptops have CPUs that run hot, and require a small fan to prevent overheating and eventual shutdown or damage. These fans are usually run with an on/off duty cycle that varies with ambient temperature, airflow, and CPU usage (e.g. playing video games vs. idling). The greater the CPU’s temperature, the greater the proportional time that the fan has to be on, and – since the fan is powered by the battery when not plugged into the wall – the shorter the battery life between charges.
Since a running laptop will be warmer than its surroundings, some of the excess heat will be removed through the laptop’s case:[ul]
[li]by conduction to the table or lap (modern laptops can feel hot!),[/li][li]by convection in passing air currents,[/li][li]and by radiation to the surroundings.[/li][/ul]Now, although it’s not a big deal at normal laptop temperatures, a black case radiates heat better than a white case. So, if all other factors are equal, having a black laptop case should ease the “cooling load” required of the fan, thus reducing the fan’s on/off duty cycle, lowering average current consumption, and prolonging battery life.
Since the effect will be tiny, I can’t be bothered to run the numbers, but my guess is that the black laptop will win out by less than 0.01%.
You probably shouldn’t even tell your friend the above, Azeotrope, since she sounds as though she might believe it’s an important factor in choosing a laptop. It’s not.
[However, color can be an issue in heat transfer calculations, which is why the most efficient heatsinks are anodized black when the increased performance justifies the higher cost.]