As always, having the facts will help. Read the front and back of your lease agreement. Call you leasing company and ask them any questions you have. With that in mind, here's my 2 cents.
Very few financial institutions will lease used cars. If you just want to keep your car, do as MyHarley suggested and buy it yourself. Here's something to keep in mind though. You either owe the leasing company the amount of the buy out (residual + a purchase fee) or the car, in the condition according to the terms of the lease (mileage, damage, etc.). If you've taken good care of the car and are interested in keeping it, buy it. If you you don't want to keep it, you may sell it, but be sure you get enough for it to pay the purchase option. Selling it is often a good choice if either the car is in really good condition and has a high resale value, or you've gone way over on mileage. In the case of going over on mileage, it may cost you less to sell the car, even at a loss, than to pay the milage penalty. If you are really upside down (purchase option exceeds the value of the car) and you don't have enought money, you may choose to go back to the dealer and trade it in. They can roll the amount you owe into your new vehicle payment. I do not recommend this as it perputates a bad debt situation. Eventually it will catch up to you and you will have much bigger problem.