Ball park figures are
12 kWh used = c. 14kWh needed from batteries (80-98% efficiency depending on inverter - 85% for small?)*
Maximum drain depends on your batteries - but let's say 40% drain -- so the total battery has to be:
14kWh / 40% = 35kWh
35,000 Wh / 12 V = 2,916Ah
If you want 12 hours overnight, then that number is probably right. If you wanted say, 24 hours, then you should factor in the fact that the solar will 'recharge' a certain amount every day --> meaning you won't drain so much / won't use as much energy.
Most people 'get over' this analysis just by buying too much -- and then replacing it well before it reaches the end of life --> engineering through excess
If you want to read more - this looks very good:
http://batteryuniversity.com/learn/article/bu_503_determining_power_deliver_by_the_ragone_plot
There's some simplified calculators here too:
http://rimstar.org/renewnrg/sizing_select_batteries_for_off_grid_solar_system.htm
To do a FULL off-grid analysis you need to consider:
1. Inverter will always draw a little bit of power from the batteries -- around 10-20W - 24 hours a day, even when you aren't using power.
2. How long you expect to be without generation -- e.g. for solar, that's about the 'probability' that it will be really cloudy for days and days at a time, or, if not well maintained your power-controller could die... That's why lots of people just chuck in a manual petrol generator, just in case... It's cheaper to design in a bit of fuel, than design-out very low probability sequences of cloudy days
3. Your inverter WILL break, after a certain amount of time... so that might be more-likely than the probability you're go for in 2 above
4. The Ragone / discharge plots above - for YOUR specific battery:
http://batteryuniversity.com/learn/article/bu_503_determining_power_deliver_by_the_ragone_plot
e.g:
http://www.sinetech.co.za/ops_batteries/ops_batteries.pdf
20% drain will mean 4200 cycles (= 11 years a 1-cycle per day),
40% drain will mean 2100 cycles (6 years)
60% drain will mean 1400 cycles (4 years?)
80% drain will mean 800 cycles (2 years)
>80% -- all bets are off!
5. Power Factor - e.g. if you're driving pumps, this can be an issue
6. How exactly you're discharging - and the related battery issues -> see battery university. Batteries 'like' certain types of loads depending on their chemistry.
* -example, this
http://e-systems.co.za/System/FabSheets/0FBI3000P.pdf is 85% efficient