Forget the one-percent rule
You'll hear that refinancing is worth it if you can drop your rate by 1%. It's a rule of thumb that ignores loan size, how long you've held the loan, and what the refinance costs. On a large balance a quarter point can be worth it; on a small one a full point might not be.
The actual test
Divide the cost of the refinance by the monthly saving. That's your break-even in months. Staying in the home comfortably past it, the refinance pays. It's the same arithmetic as paying points, and it's the only test that adapts to your numbers.
The reset nobody mentions
Refinancing a loan you're eight years into back to a fresh 30-year term lowers the payment partly because you've stretched the remaining balance over 30 years again. You can pay less monthly and still pay more interest over the life of the loan. Ask to see total interest both ways, not just the new payment.
Reasons beyond rate
Dropping mortgage insurance once you have equity, consolidating a second lien or high-rate debt, moving off an adjustable rate, shortening the term, or pulling cash out for a specific purpose. Several of these can be worth doing even when the rate barely moves.