One of the perks about being 1K on UA is that you'll get a hotel and meal voucher for an overnight stay if UA can't get you out the same day, even if if the cause is weather or other factors beyond UA's control.

If you're not 1K, then you'd have to pay unless the delay is UA's fault (mechanical or personnel issues). So it's in your best interest to try to get out the same day on another flight, whether that's on UA or another airline (though you'd obviously prefer UA if you're doing a MR). Generally the airlines are pretty nice about switching you to a new flight, and even when flying with airlines I've never had status with, I have not had to pay a fee. But I think it depends on the agent you happen to come across. As with everything, being nice and patient usually helps.

Sometimes, though, another flight just isn't available. You can minimize the risk of having a nightmare missed flight scenario by avoiding connections that could contribute to that (such as ORD in winter or MSP during summer afternoons). With a MR you'd presumably have a bit more flexibility.