Hi Kara,
Your mom has the right idea, that you should save seed from heirloom or non-hybrid seed, but she is incorrect on why.
Seed saved from Hybrids will not grow true to form. They will develop fruit, but the fruit will not look/taste like the original plant so you may end up with a crop of stuff you don't want.
Sometimes hybrids don't produce viable seed so you can't grow a crop at all. Maybe this information was sort of remembered by your mom and that's how she told you what she did.
I'm confused about the black-eyed peas though. Are you sure they are a hybrid? I'm under the impression that beans and peas, and other legumes sold for home use are not hybrids.
Remy