It's not really "stupid" of her, even if you find it irrational. Superstitions often come from a deep place in people, it's not just a matter of logic. Your mom genuinely believes in this stuff, and it means something to her. Is it scientifically justified? No, obviously not. But that doesn't change the fact that refusing to lend you her watch is her way of feeling in control of something that worries her, even if it's irrational.
The thing with superstitions is that everyone has one or two, even if you think you're completely rational. I'm not into that whole negative energy thing at all, but I get why your mom holds onto this. It's like a little bit of magic we give ourselves so we don't freak out. She just doesn't trust Friday the 13ths, that's it. It costs her nothing to not lend you her watch that day, so honestly, why are you really trying to convince her? It's not going to change her mind, you'll just end up with a frustrated mom.
If you really need her watch, ask her another day. Otherwise, let her have her superstition. Worst case, you borrow something else or figure it out without it. It's not a big deal, and it's not going to ruin your relationship just because she refuses one thing one Friday the 13th a year.