Haha yeah, superstitions are definitely something that divides people. After 13 years doing deliveries, I've seen customers who refused to receive packages on Friday the 13th, others who didn't care at all. Personally, I don't think it brings bad luck. I mean, statistically speaking, there aren't more accidents or weird stuff happening that day than any other Friday. It's mostly a psychological thing - if you believe it's gonna bring you bad luck, you're gonna be more stressed and then more clumsy, you know?
Anyway, I get your mom though. Beliefs don't work with pure logic. It's more of an emotional, cultural thing. She believes in it, so for her it's real in her head. It's not stupid on her part, it's just... different from you. I'd say it's not necessarily nonsense, it's just that not everyone lives with the same convictions.
If you really want to borrow her watch that day, you could reassure her calmly, explain that you'll take good care of it and all. But honestly? To keep a good relationship, it's probably not worth pushing on this. Superstitions exist everywhere, even with the most rational people. I've got a little routine before big winter trips for ice fishing - it's not logical, but it puts me in a good mood, you know?