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.
If your monthly payment on a fixed-rate loan includes escrow amounts for taxes and insurance, your payment each month could change over time due to changes in property taxes, insurance, or community association fees.
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.
You can also save money on your mortgage refinancing by paying "points." This is a fee that effectively lowers the interest rate of your loan. If you plan on staying in your home for a long time, this long-term strategy can be an excellent way to save thousands of dollars.