I checked with tax experts. This is not a new problem. Travel prizes are often valued by donors at full Y fares and hotel rack rates, for example. FTers, and the IRS, know that nobody pays those. So the IRS won't be surprised to hear that you value a travel-related gift lower than its donor.
Suggestions were to report the amount from the 1099-MISC so the IRS computers find a match. Then, either (depending on who you ask) either subtract your adjustment in the next line, or file an amended return with the adjustment later.
In either case, you have to explain the logic behind your valuation. That's easy for a travel award. A travel agent can document what the itinerary ought to cost. (Some will charge a fee unless you're a regular customer, but that's beside the point for this thread.) It's tougher with miles, unless you've already used them for something and can document what that ticket would have cost. Expect to be questioned. A likely outcome is that they'll suggest a compromise between your valuation and the giver's. Take it.
Added later: This is from TurboTax help for 2003 returns. TurboTax is in the "subtract the adjustment" camp. I doubt things will change much for 2004:
Fair Market Value of Prizes and Awards
If you know that the fair market value of a noncash prize or award you received is less than the value reported on Form 1099-MISC, enter an adjustment as a negative amount.
For example, let's say you received a Form 1099-MISC that shows $1,000 in box 3 for a television set you won in a radio station contest. But you know that the discount store in your city sells the same television (same brand and model) for only $750.
Because the fair market value (FMV) of the television is $750, you can enter the $250 difference as a negative adjustment on the Other Miscellaneous Income screen. Enter "PRIZE FMV ADJUSTMENT" for the description and "-250" for the amount.
It's a good idea to document your adjustments with ad clippings or similar items. Keep them with your other tax documents in case you're ever asked about them.