If you want to make sure the interest rate your lender offers you is the rate you get when you close the loan, ask about a mortgage lock-in (also called a rate lock or rate commitment). Any lock-in promise should be in writing. Make sure your lender explains any costs or obligations before you sign.
When refinancing, don't take the first offer that comes around. Instead you should actively compare both interest rates and lenders.
Pull your credit report from the bureaus and review it for any negative items (late pays, collections etc) and inaccurate detail. Try to dispute negative items and remove them from the report. If required pay off any unpaid debt. Otherwise, you won't get a low rate and may not even qualify. Of course there are lenders in the subprime market who may offer you a bad credit refinance loan, but it's better to avoid them as they'll possible charge higher rates and fees.
Be Proactive. Your loan representative should keep in contact with you even if there isn't any action on your loan. But do your part to keep the lines of communication open as well. If you don't hear from your loan representative for a few days, pick up the phone and ask for an update. The old adage that the squeaky wheel gets the grease definitely applies in this situation.
If you are able to get a lower rate that what you currently have, you can save tens of thousands of dollars over the life of your loan. Also, most lenders don't charge as many fees to refinance a mortgage and depending on how much equity you have in your home you may be able to roll the closing costs into your new loan, still have a lower balance than your original loan, a lower rate, and a lower payment.