Could a black hole absorb enough mass to become a Big Bang?

No. According to general relativity, a black hole does not have the same kind of space-time geometry as the 'initial singularity' from which the universe seems to have emerged. All you would end up with after swallowing a galaxy worth of mass would be a so-called 'supermassive' black hole weighting several billion times the mass of the Sun, and occupying a volume of space about the size of our solar system. It would then just sit there for 10^100 years or more before completely evaporating away into a plasma of electrons, positrons and so on. But so far as we know from the theory, no Big Bang.

The caviat to this is that Stephen Hawking thinks that once matter falls into a black hole and reaches the Singularity, this Singularity at the quantum scale may actually become a gateway or a spawning ground for a new universe which would exist in some adjacent set of spacetime dimensions. Black holes formed in our universe, according to Lee Smolin, may actually spawn universes beyond our own. I can't imagine any way to test such a conjecture because we can only make observations within our own spacetime.