Piazza della Pigna, or Pinecone Square, named for a very famous pinecone found there. The pinecone in question is a massive first century orginal Roman bronze (created by one Publius Cinsius Salvius), dating back to the 1st century AD. It was a fountain at the baths of Agrippa, which were, not surprisingly, located in the exact same area. Imagine the odds!
Once water gushed out of the top of the pinecone, as well as from little spouts all around the base. A few of these spouts still exist. Since the Pigna was discovered in the 8th century, the darkest of the dark ages, it would have been at risk of being melted down and used for weaponry, a fate that most ancient Roman bronzes suffered. But the pinecone, being a very spiritual symbol in Christianity, was revered, and so the sculpture was preserved.
In fact, the pinecone is a revered symbol in many religions. For Christians it represents eternal life, but it can also be seen in Egyptian, Mithraic, and other pagan art. Some scholars have suggested that because of this shared significance, the pinecone also represented the temporary peace reached between pagans and Christians in Rome when the latter changed their holy day from Saturday (Sabbath) to Sunday, to coincide with the Pagans' holy day in 321 AD. But now I'm getting way off topic...
Shortly after its discovery in the 8th century, the Pigna was given a place of honor in the center of the courtyard of Constantine's Basilica of St. Peter. There it stayed until the basilica was demolished to make way for the new St. Peter's, 500 years ago, and since then it has resided in an enormous niche in the monumental Belvedere Courtyard at the Vatican, now called the Pinecone Courtyard.
This sculpture has indeed had many things named for it. The courtyard, the piazza, the piazza's modest church San Giovanni della Pigna, the nearby Via della Pigna, and in fact, the whole neighborhood. Pigna is one of the Rome's 22 Rioni (neighborhoods). Rione Pigna is medium-sized, and streches from the Pantheon to Via del Corso, from Piazza Venezia to Largo Argentina, and its symbol is, you guessed it, a pinecone.