Go to one of the large national bidding travel websites and type in your trip information under the "Cars" heading. Because you are renting the car from a major airport, you can bid on rates that would not be available at the average rental center. With so many cars in one place, rental companies often offer specials for airport rentals.
Elect to bid your own rate for the rental once you have seen the rates offered. Click on the "Name Your Own Price" button and enter the daily rate you would like to pay. Keep in mind that this rate does not include daily taxes or fees and that the total cost will be higher than your bid. Take note of the likelihood of success of your rate as given by the site, which will help you determine whether there is any chance it will be accepted or not. If the bid is not accepted, the site will reject any further bids and you cannot book so don't go too low.
Pay for the rental with a credit card once your bid has been accepted. If using a major credit card (AMEX, Master Card, Visa), your rental insurance should be covered as long as you decline any further coverage from the rental company when offered. Double-check your card benefits to verify your coverage.
Pick up the car at LAX on the specified date and time of your reservation knowing that you had a hand in your rate and it is at least a competitive one. Because the bidding system often only uses major companies for its services, you will be assured of a company you know and a location in the airport proper for pick up and drop off.
Search each individual rental company site one by one for the best rates, then search the Internet for "rental car coupon codes," or "rental car discount codes." You will find a variety of code providers, some legitimate and many who are just trying to sell something. If any surveys or fees involved, pass on the coupon codes (see Resources 2 and 3 for two reliable options).
Plug in the coupon and discount codes applicable to your rental request and see what happens. At times you will be given discounts off the rate, upgraded car categories or the like. At other times you will have nothing to show for the effort, but rates directly from the company sometimes do beat the mass marketing travel sites so it is worth a look anyway.
After securing the best rate, book the car and follow through with the reservation as you normally would.