Iida is the name of a family that lived in the area at the time Tokugawa Ieyasu set up his new capital in Edo. Ieyasu appointed a certain 飯田喜兵衛 Iida Kihei to the position of village headman, and the area soon came to be known as 飯田町 Iidamachi Iida Town. The bridge over Edo Castle’s outer moat was named 飯田橋 Iidabashi Iida Bridge. If you want to go into detail about the etymology of this area, please see my article here.
The station gives you access, though not directly, to 小石川後楽園 Koishikawa Kōrakuen, the stunning gardens of the 水戸徳川家 Mito Tokugawa-ke lords of Mito Domain. The palace of the lords of Mito included this garden, but also included present day 東京ドーム Tōkyō Dōmu Tōkyō Dome, which is located next to the garden.
If you make the walk to the garden and to Tōkyō Dome, I suggest walking a little farther to 水道橋 Suidōbashi where the shōgunate used to have an elevated aqueduct – one of the greatest engineering marvels of Pre-Modern Japan. And while you’re at it, just walk a little further to the 東京都水道歴史館 Tōkyō-to Suidō Rekishikan Tōkyō Water Works Museum. The museum teaches you all about wells, aqueducts, and sewer systems from the Edo Period to present day. It may sound boring, but trust me. It’s one of the coolest Japanese history museums I’ve ever been to.
They don’t have an English website, but this webpage may help. If you’re a fan of this blog, I’m pretty sure you’ll love the museum.
Oh, also, Iidabashi station is probably one of the best access points for Kagurazaka.
By the way if you compare this picture with the top picture, they are both taken from the 東京理科大学 Tōkyō Rika Daigaku Tōkyō University of Science which was founded in 1881. The western style house in the top picture was was the 逓信総合博物館 Teishin Sōgō-Hakubutsukan Museum of Communications, which is now located in Ōtemachi.