If you are refinancing from one ARM to another, check the initial rate and the fully-indexed rate. Also ask about the rate adjustments you might face over the term of the loan.
Ask the company carrying your current title insurance policy what it would cost to reissue the policy for a new loan. This may reduce your cost.
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.
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.