He had no wife or children to leave his money to.
As I recall The Smithsonian was actually founded with with money that congress apportioned to replace the money he gifted. Because the language behind gifting was so vague they kept dicking around and eventually invested his money in some gold scam and lost it all.
He actually left his money to his nephew, but when his nephew died without heirs six years later the money reverted to the US government.
Although he never visited the US during his lifetime, his remains are there now. He, or rather his tomb, is actually on display at the Smithsonian, where he was re-interred after his death in Italy. (I believe his casket is actually buried below the monument.)