Searching through older topics on C&F, I discovered this topic from 2006: http://forums.chiffandfipple.com/viewtopic.php?f=1&t=37741&start=0
I’ve been doing a little research, and discovered the statue is the work of Italian sculptor Pasquale Romanelli, (Florence, 1812-1887)
It seems there are actually a number of identical sculptures in existence, dating from 1863 and carved from White Carrara marble (whether they are all carved by Romanelli himself or by people in his workshop is unclear)
As well as the one in the library of the Union League, there is also a version of the sculpture in the Barr Branch Library at 1701 South Jefferson Avenue and Lafayette Street, St Louis; and a further one in Newark Public Library, Washington Street, Newark, New Jersey:

“Benjamin Franklin and His Whistle” Newark Library
White Carrara marble - carved, in Italy, by Pasquale Romanelli, in 1863
Interestingly, of the three, only the Newark one actually has the title “Benjamin Franklin and his Whistle” engraved around the base.
And then there is another example of the sculpture - in the Gallleria d’Arte Moderna, located in the Villa Saluzzo in Nervi, Genoa, Italy. This particular example has the Inventory no. G.A.M. 941

White Carrara marble - carved, in Italy, by Pasquale Romanelli, in 1863
Originally from the collection of the Prince Odone of Savoy, Duke of Montferrat, this latter sculpture was given to the town of Genoa by his father, King Victor Emmanuel II in 1866 after Odone’s death
However this particular sculpture is entitled: “Beethoven Giovinetto” (Young Beethoven) …