When NASA no longer has a need for an object -- be it a spacesuit, a piece of equipment, or even an entire spacecraft! -- the object usually goes to a museum. It seems that the Smithsonian Institution (particularly its National Air and Space Museum) usually has the "best" copy (or sometimes only copy) of a particular type of object, either on display or in its collections.
Is there an official policy that the Smithsonian has priority over other possible venues for NASA artifacts?
At least in some cases, yes.
The Smithsonian was selected to receive one of the flown shuttle Orbiters before the plan for figuring out where to send the flown shuttle Orbiters was even written.
The NASA Authorization Act of 2008 (P.L. 110-422) directs NASA to “submit to Congress a plan describing the process for the disposition of the remaining Orbiters and other Space Shuttle program-related hardware after the retirement of the Space Shuttle fleet.” NASA advised the Congress that it would begin discussions with the Smithsonian Institution, National Air and Space Museum regarding accession of a flown Space Shuttle Orbiter to the national collection. The National Air and Space Museum houses the national collection of aerospace artifacts. NASA also advised the Congress that the Agency would issue an RFI to gauge the level and scope of interest of U.S. organizations in acquiring the two (2) other Orbiters and other major flight hardware (such as the SSMEs) for public display once NASA’s programmatic requirements for the assets have been satisfied.
From Space Shuttle Program Request for Information (emphasis mine)
Also from Space Shuttle Program Artifacts Pamphlet
When will a decision be made pertaining to who will receive one of the Shuttle orbiters at the conclusion of the program?
NASA’s primary focus is to ensure that the Space Shuttle safely and successfully completes its mission of finishing the assembly of the International Space Station by the end of 2010. NASA intends to transfer the Shuttle Orbiter Discovery to the Smithsonian Institution, National Air and Space Museum. NASA has made no decisions on final disposition of the Shuttle orbiters Atlantis and Endeavour.
