Hmph. For some reason it is very difficult to find the Amp-Hour ratings for automotive batteries online. I'm gonna say 150 Amp-Hours, but take it with a grain of salt, as it is only a rough guesstimate.
If you assume 100 watts for the laptop (including losses in the inverter), 150 Amp-Hours would give you about 18 hours of runtime. However, deep-cycling a cheap car battery is going to ruin it eventually. I suppose if you only need it to last 3 weeks, that doesn't really matter.
Depending on how much you use the computer, you may need to drag it across the way to your uncle's every couple of days to charge it. And you'll be without power while it charges, so you'll need two (so that you can have one always hooked up to the charger, and swap them when the one you're using dies).
The extension cord idea is sounding better all the time.