Cosmologists assume that the universe is homogeneous on large scales and, in particular, that the unobservable parts are similar to the observable universe. As I understand it, one important reason for this assumption is simply something along the lines of “hey, it would be really strange and surprising if our corner of the universe turned out to be special.” While this does seem very reasonable to me, I was wondering what other reasons (if any) we have for the assumption. Is it, for instance, predicted by the big bang model?
In the standard big bang model, homogeneity is an assumption - therefore it cannot predict homogeneity. Homogeneity is not a requirement of the model and indeed you can build "inhomogeneous big bang models" - though they must still be consistent with the observational evidence of a high degree of homogeneity on large scales within the observable universe.
The obvious observational evidence for homogeneity on large scales (and of course, in the observable universe) is the smoothness of the cosmic microwave background (CMB). The CMB is very, very uniform - with perturbations (bar the large-scale dipole) at the level of 1 part in $10^5$.
We can say very little about the unobservable universe, since it is... unobservable. And because it is unobservable, it has no influence on us and cannot affect the utility of the models we use to explain the observable universe.
NB: the big bang is just a model of how the universe works, with adjustable parameters including dark matter, vacuum energy density and homogeneity, not a theory.