The ideal goal of a student loan refinancing is to lock in a lower interest rate. The lower rate ensures that you pay less over time, and it lowers the rate of your payments quickly as well. When you're given a refinancing rate by a lender, compare that rate to the rate that you're currently paying. Also check to see if it's an introductory rate, a variable rate or a locked in rate. Introductory periods end, variable rates change with the market, and a locked in rate stays constant.
If you cannot get a lower rate interest rate when you refinance, you can often extend the time of the loan. This means that you make smaller payments over a longer period of time, but your interest keeps accruing as you do. You'll end up paying more in the long run, but you won't feel the burden as much from month to month. Alternatively, you may be given a lower interest rate when you refinance, but the loan will be extended much further as part of the deal. Do the math and work out how much you're going to pay over the life of the loan to see whether the lower refinance rate is the best one.
If you have multiple student loans, which many recent graduates do, then you can consolidate those loans into a single monthly payment. This single payment will likely be larger than any individual loan payment, but it will be less than paying each loan separately. The rate of interest on this form of refinancing may look good, but you also need to work out whether what you're paying on the single bulk loan costs you more in the long run than the interest on the smaller loans. Only once you've run the numbers on the new, consolidated loan and the interest that you're paying can you see how it measures up to your other options.
Your interest rate is an important consideration when you're looking at your refinancing options, but it isn't the only one. You also have to look at whether your loan is being extended for a longer period of time and whether you lose benefits such as a grace period. Other loans, particularly those refinanced by the federal government, will raise your payments over time on the assumption that you will have become gainfully employed and can afford higher payments down the line. While a lower interest rate is always good, you have to consider it as part of a whole to see whether or not the rate that you're getting is really saving you money.