Here are some of their names.
11 December 1951, New York Times, pg. 35:
The stone giants, which the late Mayor Fiorello H. La Guardia nicknamed Patience and Fortitude and which library workers call Lord Lenox and Lady Astor, or Leo and Leonora, emerged with handsome collars of cedar rope with lights for after-dark effect and huge red bows.
13 May 1955, New York Times, pg. 27:
The carved cats never got official names, except that Mayor LaGuardia once called them "Patience" and "Fortitude."
30 June 1960, New York Times, pg. 31:
The hurt lion is sometimes known as the uptown lion, or the north lion, to differentiate him from his twin on the southern edge of the main steps. They've also been called Lord Lenox and Lady Astor, and Leo and Leonora, but these names are at least half wrong. Both lions are males.
16 April 1961, New York Times, pg. BR8:
The lions have had a couple of dedications, one of a children's book by James Faugherty reading, "To Lady Astor and Lord Lenox, the Library Lions..."
28 March 1969, New York Times, pg. 25 ad:
Leo and Leonora, Lord Lenox and Lady Astor, Patience and Fortitude are some of the nicknames New Yorkers have given the heroic lions who guard the entrance to New York's Public Library at Fifth Avenue and 42nd. Just five blocks up Fifth, you'll find perfect 7" Alvastone replicas of the lions. These bookend lions are approved by the Library, and are on sale in our Sculpture Gallery. You can own them for $17.50 the pair, and give them your own nickname.
5 November 1975, New York Times, pg. 42:
In that most attractive of children's books about New York, "Nicholas: A Manhattan Christmas Story," she (Anne Carroll Moore-ed.) says that the first library lion is named Leo Astor and the second Leo Lenox.