The first possibly substantiated rumor has surfaced with a photo reportedly of the iPhone 5S' motherboard. In the picture, there is a chip clearly marked "A7." This chip should be the successor to the current A6 processor used in the iPhone and iPad 4. iOSDoc claims that this A7 chip is a 1.2GHz quad-core part.
The new chip is said to be paired with a PowerVR quad-core SGX554MP4 graphics chip and 2GB of dual-channel LPDDR-800. The RAM is an upgrade from the current 1GB of LPDDR2-1066. The device is also said to come with iOS 7, which makes sense, and the upgraded OS is set to feature many cool new features.
Of course, we've heard lots of rumors about Apple, with some saying that there are three phones in the works by Apple. I'd continue to take rumors with a grain of salt, though this one has a bit more credibility due to the picture.