This is an example of a standard issue U.S. Marine Field Cover or boonie cap. It has the MARPAT or Marine Pattern digital desert camouflage.
This cap was worn by the donor Capt. Matthew Dwyer. He was the first 1st Lieutenant selected by the Marine Corps to fly the MV-22 out of Marine Squadron VT-2 in Milton, Florida. He then received V-22 flight training in VMMT-204 in August 2006 and was assigned to VMM-162, the second operational V-22 squadron. He was then assigned as the Aviation Life Support Systems officer and deployed to Al Asad Air base in Iraq March-October 2008. He had the privilege of flying Barack Obama, then the Democratic presidential candidate, from Ar Ramadi, Iraq to Amman, Jordan, a flight only capable in a V-22 aircraft. Following his deployment and flying numerous combat missions in the Middle East, he was then deployed with VMM-162 (reinforced) as part of the 24th Marine Expeditionary Unit in 2010 to Haiti for humanitarian assistance and disaster relief after the earthquake.
This object is not on display at the National Air and Space Museum. It is either on loan or in storage.