Okay, I know the first thing that popped into your head was the iPhone robot voice. But, though that acronym is great and all, it might have ruined Siri as a name.
The name Siri may have started as a reverse of the name Iris. It's fresh, cute, modern, and great twin names.
Something else interesting about the name Siri is the character Siri Tachi from the Star Wars universe. She was a spunky blond human girl who wielded a violet lightsaber. She was close friends with Obi-Wan Kenobi during their Padawan years. While on a mission not long before the Star Wars Ep 1 time period, they realize their love for one another. Since love is forbidden for Jedi, and the mission was already pretty complicated as it was, they decided to wait until afterwards to figure out what to do about. Tragically, Siri was killed while bravely protecting the rest of those on the mission.
Anyway, pretty fun stuff. However, as our iPhone robot friend coincidentally has the same name, it may be a while before we can use it.