What triggers the eye of the herald opening?
According to the League of Legends wikia (emphasis mine),
After being in combat for three seconds, the eye on Rift Herald's back will open. Champion basic attacks against the eye deal (12% of Rift Herald's maximum health) bonus true damage, causing the eye to close for 10 seconds, reduced by 2.5 seconds every time she is struck by a champion's basic attack.
From the LoL wiki:
After being in combat for three seconds, the eye on Rift Herald's back will open. Champion basic attacks against the eye deal (12% of Rift Herald's maximum health) bonus true damage, causing the eye to close for 10 seconds, reduced by 2.5 seconds every time she is struck by a champion's basic attack.