Okay, first things first, your ex is kind of a nut for even giving this any credibility to begin with.
Second, I can't pretend to know what's going on with your son, having never met him. I also don't know you personally or your ex, so take what I say with a grain of salt and keep that in mind. But I can give you some professional insight on what is common in kids:
First, keep in mind that this is a normal stage for kids around that age and it can happen for a lot of different reasons. You are smart to want to nip it in the bud now, but don't panic worrying that he's going to grow up to be a sociopath or anything...this is (as much as anyone is) normal! But it's a pain in the butt when you're the mom.
Kid will lie for a lot of reasons...the main two are attention and control. Attention...he wants it, and even negative attention is attention (despite the fact that you give him positive attention too). Sometimes they can't get enough. And at 9, he may not want to ask for it because it's not a big boy thing to do.
Second...the control issue. They all have it. One aspect of this is to see how much he can get away with...testing his boundaries for the preteen and teen years. If you are a pushover now, you bet he'll be storing that information for later use. Also, if he's upset about the divorce (and maybe his dad not spending as much time with him), it could be that this is a way to get Dad involved, to get Dad talking to you, to get Dad paying attention to him, and to feel like he has some sort of control over something. He couldn't control the divorce, he can't control his dad's behavior or attitude toward work, but he sure can control how he manipulates with his lies. Maybe that's all he feels he's in charge of right now...what comes out of his mouth when someone asks a question.
You might want to take a look at whether he lies at school, about homework, to friends, etc., also, which might give you a little more of a clue as to what's behind it. Most of the time, lies are to stay out of trouble and kids just don't get that it would be easier to (not eat all the granola bars, brush teeth when you're supposed to, fill-in-the-blank) than to lie and cover it up. Infuriatingly, they just don't get it.
My best advice would be to acknowledge it but to keep it low key. The more you feed it, the more it will happen, but you can't have him thinking you are Clueless Mom, either. Some kids simply do it for the thrill of seeing if they can get away with it; if you treat it as no big deal, that will take away the thrill some. You might also try asking him how he thinks you feel when he lies to you to get him to see the other point of view.
And now that you don't have to be perfect, you can be good.