Again, I want to note that there is wisdom in this bit of advice. Insults can't hurt you if you don't let them. However, it's also worth noting that we don't always have control over how this stuff affects us. I mean, we can certainly control to a decent extent how much we let on the impact of insults and slurs. If you call me something disparaging, it's quite possible that it'll roll right off me because, after all, you're just some random person I don't even know. Rationally, I know there's no reason to be concerned by this interaction. But we're not purely rational beasts.
I'm not saying this is true, but in scenarios like the above with your mom, have you considered that she really wasn't as immune to the hurtful words as she let on? Maybe she really was impervious to such attacks because, after all, life is indeed too short. But it's also possible that it hurt each and every time someone made fun of her because of her weight, but she knew that being hurt made her appear weak.
I'm not saying that this is the case for your mom in particular, but I think people put up these kinds of facades all the time. And while it's true that ignoring such comments or trying to have thicker skin may be one of the more effective strategies, I think it does a disservice when people that do honestly call out hurtful speech are portrayed as overly sensitive as opposed to acknowledging the power of hurtful words.