According to the website at the Museo di Storia della Scienza Galileo’s finger was detached by Anton Francesco Gori on March 12, 1737, when Galileo’s remains were moved from the original grave to the monumental tomb at Basilica di Santa Croce in Florence. The finger became the property of Angelo Maria Bandini and was long exhibited at the Biblioteca Laurenziana. In 1841, the relic was transferred to the just-opened Tribuna di Galileo in the Museo di Fisica e Storia Naturale. 
Together with the Medici-Lorraine instruments, it was eventually moved to the Museo di Storia della Scienza in 1927. On the marble base is carved a commemorative inscription by Tommaso Perelli.
I have always been fascinated by the fact that Galileo’s finger has been preserved. I mean, to what end? Awe? Inspiration? Fanatacism?
I believe there was, at the time, a fascination with collecting the bones of saints (essentialism) but I’m guessing that issues related to the information below played a role in the selection of the particular bone displayed.
From the History of the Longest Finger at:
http://www.ooze.com/finger/html/history.html
“Giving someone ‘the finger’ is one of the basest violations in modern culture, but its origins date back over 2500 years. The first written record of the insult occurred in ancient Greece, where the playwright Aristophanes (the Adam Sandler of his day) made a crude joke mixing up the middle finger and the penis. Even back then, the bird was considered an aggressive, phallic put-down.
It has been argued by anthropologists that the finger is a a variant of a classic ‘phallic aggressive’ gesture used by primates. By jabbing a threatening phallus at your enemy like a wild animal, you aren’t just belittling him, but also making him your sexual inferior. Instead of using a real penis, civilized Janes and Platos called upon the substitute wieners within their own hands to mock, threaten, and humiliate opponents.
And boy, did it. When the Romans imported the art, music, and culture of the Greeks, the finger came along, too. Roman Emperor Caligula, a pioneer in perversity, frequently shocked his citizens by forcing them to kiss his middle finger instead of his hand. One of his subjects, Cassius, who Caligula often taunted as being too effeminate, finally had enough humiliation and assassinated him. Clearly, the bird was not to be taken lightly.
During the Middle Ages, the finger went underground. It was still known, but the Catholic Church frowned upon its use, as the middle finger was supposed to be holy in the Mass. The unholy insult lurked deep within the hearts of filthy- minded folks everywhere, hiding from sight until the 19th century when it began to crop up again thanks to a new invention -photography.”
I put an inquiry into Bruce for his spin on this great question. Thanks!!!
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Florence Museum’s Display of Remains Reflects Latest Chapter in Galileo’s Legacy – http://nyti.ms/bW3TXO – New York Times article puts a little historical perspective on this.