I disagree with A LOT of what others have said here. I am ASTONISHED that only one other person mentioned what I'm about to say.
Additional security deposit.
Lets face it people, bad credit and renters go hand and hand. It doesn't make someone a bad person, just means they went through bad times. Yes he has a criminal history, the good news is that it was from 4 years ago and you didn't find a whole big pile of new charges. It's possible he has gotten his act together but because he has this stuff in his past, they know they won't get approved to live most places. So yes, she did try to play you out to some degree but she is kind of trying to do right by telling you. This guy could have just started coming over all the time and basically moved in and them not tell you. So why not simply say, he can move in but he needs to come up with an additional 1 month rent as security deposit. That way, if **** hits the fan you've got that extra months security in hand should they stop paying. You say he's got a steady paying job making decent money so this shouldn't be a problem. After one year, if they are causing problems, you can simply decline to renew the lease.
Want to get really bold/creative? up the asking price on the rent as well. tell them you want $100 more per month to add him on the lease.