I've never actually encountered this problem in a lab so I can't say for sure. The usual sophomore lab experiment involves using an acid, an amine base and another relatively non polar compound that is neither basic nor acidic. Addition of dilute acid could surely be used to pull the amine phase into water in this case.
You're on the right track with using base to put the acid into water. At that point we're dealing with Benzil vs. Dibromobenzene. There is a significant difference in polarity between these two compounds. A reasonable suggestion would be combining a very non polar solvent (Perhaps diethyl ether) with a more polar organic solvent such as DMSO or DMF. Dibromobenzene might go into the diethyl ether, while benzil might prefer the more polar solvent. And I'm pretty sure DMSO and Diethylether do not mix due to their very different polarities. I don't know for sure what solvent combination would be best, though.
The other possibility would be that the ketone portion of benzil might form a stable protonated product like an amine, in which case you could pull it into water with dilute acid.
Sorry if you knew this already and just wanted a specific, definite answer. Those are just educated guesses.