Letters of Marque are issued to private entities. The documents allow a privately owned vessel to attack specifically named enemies of the issuing nation on the high seas. This was a way of dealing with pirates in the 1700's. The receivers of the Letters of Marque are typically called privateers.
There is legal grounds for the United States to issue these. The U.S. Constitution gives Congress the power to issue Letters of Marque in Article I, Section 8. As much as our Congress likes to spend money, they may consider this idea. Secondly, those nations that signed the Declaration of Paris in 1856 agreed to no longer issue Letters of Marque. The United States is not a signer of the Declaration of Paris. Finally, we have used Letters of Marque as recently as World War Two. Admittedly, that is the only time the U.S government has since the War of 1812, but we have done it.
Historically, a privateer would pay a fee to a nation to get a Letter of Marque. Let's say $100,000 in today's world. The privateer would be responsible for outfitting and training their crews, and would be entitled to a percentage of any "booty" captured under the Letters (let's call it 85%). The United States would designate specific "hostile forces", such as pirates operating off the coast of Somalia, and would get the 15% "booty" not accounted for in the last sentence. One last note, the privateers would also be able to deal out punishment to any pirates they captured. The Letters are very specific in the areas and enemies privateers are allowed to operate. It would not be a blank check to cause mayhem on the high seas.
I think this is a very smart way to handle the pirates. The United States doesn't have to beef up it's military presence near Somalia, nor do we put additional funds and resources into the area. Private companies or individuals would go forth and deal with the pirates. Already the pirates have said they are going to retaliate on future U.S. vessels they capture, and they have tried as recently as today. Letters of Marque were quite effective against pirates in the past, and can be once again.