The 1851 restrike dollar struck in copper. These were deliberately struck for sale to collectors. Early 2001 articles in Coin World by Dave Bowers discuss practices by the mint in those years.
Silver restrikes of this date were known as early as lot 134 of Cogan's May 1859 Simon Gratz sale but the earliest occurrence for one of these copper pieces appears to be lot 1995 of Cogan's October 1865 F.S. Edwards sale which may indicate that these were struck a few years later, either 1863 or 1865, with another batch of silver examples.
In copper, about a dozen examples are believed to exist of which Virgil Brand owned at least 4 (journal #s 18146, 40712, 44166 and 82007).
There is also a single example known struck in nickel J133/P160 formerly from the Newcomer and Farouk collections. Since nickel wasn't really used on dollar patterns until 1869, this may have been struck along with a third batch of silver restrikes circa 1869-1871, or may be a plated copper piece.
Additional information can be found in "Silver Dollars and Trade Dollars of the United States" by Dave Bowers and Mark Borckardt.
It is unclear if these were struck from multiple reverse dies. The mint had at least 2 "No Motto" reverse dies on hand in the 1870s.
Photo courtesy of American Numismatic Rarities.