There are a lot of terms you will have to define to make your experiment have merit. It's difficult to find very good technical datasheets for batteries, even on the web. Many years ago, Eveready published a very good battery handbook, but I can't seem to find a web reference. I have attached a link to a Kodak site where they show some specs for their triple-A battery. This might be your starting point.
http://www.kodak.com/eknec/documents/e5 ... e5/K3A.pdf
This data sheet says that the capacity of their K3A battery is 1250 mAh (milliamp hours) when the battery is discharging into a 75 ohm load. That is a load of 20 mA when the battery is fresh, at 1.5 volts. It also specifies 0.8 volts as the cutoff point that defined the battery's 1250 mAh capacity. Obviously there is still energy in the battery (more than 1250 mAh), but the battery probably wouldn't appropriately power it's application lower than the 0.8 volt terminal voltage, so they define that as an arbitrary value, think of it as the useful capability of the battery.
Already you can see some of the definition problems. What constitutes the point when the bulb is still lit? It may grow dim, but visible over a wide voltage range. The battery could still be supplying current to the light bulb but it would appear dark. You could use the same criterion, the time it takes to reach the 0.8 volt minimum measured across the light bulb. Rather than a light bulb, you might consider using resistors in the experiment, and measure voltages rather than just observe whether the light is still lit. You might also discover that incandescent light bulbs present a non-linear load on the battery, that is, the effective resistance changes with filament temperature. With resistors you control that variable.
And, speaking of variables, there are several. One is discharge rate. The battery will exhibit a higher or lower capacity (than say the nominal 1250mAh) depending on the discharge rate. A high current drain, say 200 mA taken to 0.8 volts will show much lower capacity than the 1250 mAh value that the manufacturer specified with the 75 ohm load. A very low discharge rate, say 5 mA might show a much higher capacity than the data sheet shows. While a capacity of 1250 mAh suggests that you could supply 125 mA for 10 hours, or conversely 1.25 mA for 1000 hours, those are theoretical values and will not be achieved. This could become another possible hypothesis for a project.
There are miniature light bulbs that are designed for single cell flashlights that you could obtain. It also sounds like access to a digital voltmeter and milliammeter is essential. There are several experiments you could create from this starting point. You will probably learn a lot more about batteries as you do some background work. Good luck, have fun.
Rick Marz