You’ve learned a hard lesson in the importance of honesty. Lies tend to fester. Instead of making things easier, they ultimately make things so much harder. Trust is the basis of any good relationship. It is a fragile thing. Once broken, it takes heart-felt apology and commitment to being truthful — no matter how hard it is — to regain it.
Although you didn’t state so clearly, I’m guessing that you told your mom that the problems in your relationship with your boyfriend were due to him cheating, not you. I can understand why he wants to “clear his name”. To stay in relationship with you while the lie continues means that he has to always be worried about what your folks think of him. To expect him to be always seen as someone who will betray their daughter is too much to ask of him. That may well cause you to lose the relationship. It would certainly make his relationship with your parents always difficult.
The only way out is to come clean. Yes, your mom will probably be disappointed in you. Yes, she may well be angry that you threw your boyfriend under the bus. She may worry about your character and your maturity. As difficult as that all is, it’s better than leaving her with the impression that your boyfriend doesn’t deserve you and that you’d be better off without him.
Take your lumps. It’s better for everyone’s relationship with everyone else to deal with the truth. Apologize to your boyfriend. Apologize to your mother. Tell the truth. And learn the lesson this experience has taught you. Lying didn’t help. It rarely does. Instead of having one problem, you ended up with two: The problem of your cheating and the problem of the lie.
Facing up to the lie is a step toward more maturity. It will give your relationship with your parents and your boyfriend a fresh start.
I wish you well.