How many volts of AC do I need such that the sound is audible on an 8 ohm 1 watt speaker?

All speakers have an efficiency rating that is expressed in decibels(dB). A typical rating might look something like this: 90dB/1W/1M. This means that given a 1 watt input(2.83 volts@8ohm) the speaker will produce a sound pressure level(SPL) of 90 decibels while measured at a distance of 1 meter. This measurement is performed in a special type of soundproof room called an anechoic chamber. So the answer to your question depends on what the efficiency of the speaker in question is. The decibel, in regards to sound, is NOT an absolute measure but rather a logarithmic unit indicating the ratio of a physical quantity relative to a reference level. In other words, how loud one thing is when compared to another. The human threshold of hearing is 0dB. For every 3dB increase in loudness you need twice the amount of power(watts) which in turn is barely perceptable by the human ear. An increase of 10dB is necessary in order for the human ear to "perceive" twice as loud since we do not hear in a linear fashion. A whisper quiet library is around 30dB. So what all this means is that in order to play a 90dB efficiency speaker at 80dB you only need around 1/10th of 1 watt of power. This translates to just millivolts of alternating current(AC). So you can figure that for it to become just audible requires even less. As the loudness increases the power required increases exponentially.