Saints Row features a day and night cycle, though sometimes you may wish to change the time of day. This page of the guide explains how to change the in-game time.
Change the time of day
Saints Row features a 24-hour day cycle, though both the day and night are relatively long. This may be problematic if the current time of day isn't to your liking.
Fortunately, changing it is easy - you must visit your HQ (for most of the game it's the church) and interact with the mattress. This automatically skips a few hours forward - you may wake up at sunrise, midday, evening or in the middle of the night. Interact with the mattress a few times in a row if you wish to keep skipping forward.
The game doesn't let you change the weather. If there's dense fog or rainfall you want to get rid of, you can only do so by sleeping.
Day or night - does it affect anything?
The game doesn't restrict any of the activities with the time of day. You don't ever need to wait for a specific time to complete a task.
The collectibles, however, may be easier to find at night. They can stand out from the environment more, being easier to spot.