When the player first starts the game, they will be prompted to set the time on the wall clock located in the player character's room. The time of day controls the tide in Shoal Cave and the evolution of Eevee into Espeon or Umbreon. Some events are only available once a day, becoming available again after the clock passes midnight. This page provides information about the events that are tied to the clock.
These events require the start of a new day, but do not require a specific time of day.
Several people in the overworld give the player Berries once a day when spoken to.
A small handful of limited-time events will be available for one day only. These events are announced a few days in advance on Hoenn TV, and there will be broadcasts on subsequent days telling the player how many days are left until the event is active. The following events can be seen on the TV:
After defeating the Elite Four, engaging in a battle has approximately a 0.5% chance to start a swarm. The following day, the player must watch TV to learn about the swarm, which will activate it. Once activated, it will last until the end of the day after the activation occurs.
Berry growth is tied to the number of hours that pass after planting the Berry. This Bulbapedia page lists the number of hours required to grow a Berry tree. Watering a planted Berry does not impact growth time.
Shoal Cave's tide varies between high and low depending on the time of day. The different tides give the player access to different areas of the cave. High tide is from 9:00-2:59 (AM and PM), and low tide is from 3:00-8:59 (AM and PM). It is possible the intention here is for high tide to correspond to when the hour hand is 'high' on the clock and low tide is when the hand is 'low'.
If an Eevee is traded into Ruby, Sapphire, or Emerald, it can evolve into Espeon between 12:00 PM-11:59 PM (12:00-23:59) and Umbreon between 12:00 AM-11:59 AM (0:00-11:59).