This page details bugs of Persona 4.
Add a better screenshot, detail ways in which the eye has been displayed in-game, and verify the file is also in the Japanese release.
In the North American release of Persona 4, a mysterious blinking eye symbol can appear on the Persona status screen for certain characters. It is unknown exactly how the eye appears, but its appearance is the result of a bug. The most commonly seen instance is on Yosuke's Susano-O Persona. The symbol doesn't seem to have any additional effects, and appears to simply be cosmetic.
This is most of the eye itself, along with other UI pieces in the vicinity. The blinking pupil is the circle on the bottom left. All UI elements contain tags internally, and the eye is no exception. The pupil and main shape are tagged as follows:
These translate to "Awakening" and "Awakening-Eye", respectively, suggesting that the eye was used to signify the awakened second form of a character's Persona. However, there is no additional evidence to support this theory.