Bring a highlighter and some post it notes. Mark it up to emphasize the the horrible parts. The person to actually open it is likely already a Christian. Perhaps you could get someone to question their beliefs if they focus on the rape, murder, slavery, etc. featured in their "good book".
While I love the idea, do most people who call themselves Christian actually read the Bible? And those that do are likely too hardcore to be swayed by anything, and they either carry around their own Bible or use an app.
It might help someone curious, though. Hell, just trying to read the Bible cover to cover as a teen was enough to put me off for good.