Can the collision of asteroids form another moon? In the lifetime of the earth, is it possible for sufficient collisions to occur in the asteroid belt to form a prototype moon?
 A: Not really, for several reasons. 
The total mass of the asteroid belt is somewhere between $2.8\times 10^{21}$ to $3.2\times 10^{21}$ kg, about 4% of the mass of the Moon which is $7.3 × 10^{22}$ kg. 
Now, we might accept a tiny moon as a moon: consider Phobos (mass $1.07 × 10^{16}$ kg) or the outer moons of Saturn (even less massive). But the asteroid belt is also unlikely to coalesce into a moon. Obviously it has not done so over the past 4.5 billion years of the solar system, and is hence relatively unlikely to do so in the future unless conditions change. The total mass is so small compared to the planets that self-gravity is unlikely to overcome their influence (many of the asteroid families are locked in orbital resonances with Jupiter). 
Many asteroid families are also believed to be due to the breakup of larger objects, where the largest current members are partially reconstituted. Generally colliding asteroids need to have low relative velocities to merge rather than fragment each other, and this makes it very hard for asteroids on orbits of different eccentricity and inclination to meet up for a merger.
Finally, a totally semantic reason: moons orbit planets. Were the asteroid belt to merge into a single mass it would be a planet by the IAU definition. It would orbit the sun, it would be roughly spherical, and it would have cleared the neighborhood of other bodies. So it would not be a moon. 
