NOTE that this is not the same as the recently-established Cold War Recognition Certificate, which can be awarded to living veterans of the "Cold War" era, 1945 to 1991.
Reproduced below is a slightly-modified copy of a Veterans Administration information sheet on the memorial certificate. You can also view the VA web page that describes this program, and includes a nice illustration of the certificate itself.
PRESIDENTIAL MEMORIAL CERTIFICATE PROGRAM
DEPARTMENT OF VETERANS AFFAIRS
NATIONAL CEMETERY ADMINISTRATION
The PRESIDENTIAL MEMORIAL CERTIFICATE program was initiated in March 1962 by President John F. Kennedy to honor the memory of honorably discharged, deceased veterans, and has been continued by all subsequent Presidents. Statutory authority for the program is Section 112, Title 38, of the United States Code.
The United States Department of Veterans Affairs (VA) administers the program by preparing the certificate which bears the President's signature and expresses the country's grateful recognition of the veterans service in the United States Armed Forces.
ELIGIBILITYEligible recipients include the next of kin, other relatives or friends of deceased veterans. The award of a certificate to one eligible recipient does not preclude issuance to another eligible recipient.
APPLICATIONEligible recipients, or someone acting on their behalf, may apply for a Presidential Memorial Certificate in person or through the mail with any VA Regional Office or by writing a letter (no application form is required) to:
You may also call: 202-565-4964.
Applicants should provide a copy of any document which would help establish honorable military service of the deceased veteran.
In most cases involving a recent deaths, the local VA Regional Office originates the letter for a Presidential Memorial Certificate without a request from the next of kin. VA Regional Offices normally are aware of veteran deaths when processing a claim or a change in veteran benefits.
Requests for additional, replacement, or corrected certificates may be made by writing to the Director, Memorial Programs Service (403A3), National Cemetery Administration, U.S. Department of Veterans Affairs, 810 Vermont Avenue NW, Washington, DC 20420.