If you are making payments on a long term loan, say, 30 year mortgage for the past 10 to 20 years, then refinancing to another 30 year loan will not be a good option as it may increase your overall payment.
The length of time that you expect to keep the mortgage helps you determine whether it is worthwhile to pay points up front to reduce your interest rate. Unlike points paid on your original mortgage, points paid to refinance may not be fully deductible on your income taxes in the year they are paid.
Give yourself plenty of time to close. With most refinancings, your file is turned over to a closing or title company, which dictates the closing details. Like the lenders themselves, these firms are swamped when interest rates are low. Moreover, appraisers get backed up and can be difficult to schedule. So don't expect the closing to happen as quickly as anyone promises.
There is a refinancing myth that says you should not refinance your mortgage unless your interest rate will be at least two points less. This myth is not necessarily true if there are other benefits to the refinance or other reasons behind it.
Make sure that your original mortgage does not have a pre-payment penalty or early payoff penalty of any kind. Sometimes people will get into their mortgage with the mortgage having a pre-payment penalty and they will not even know about it. Pre-payment penalties usually range from 6 months to 3 years with a penalty for an early payoff. The penalty is usually about the amount of 6 months worth of your mortgage loan interest, but this varies. You would have to be able to have some significant payment and interest savings on your refinance loan to justify refinancing a mortgage loan with a pre-payment penalty.