This sounds like it could be a subconjuntival hemorrhage. A subconjunctival hemorrhage occurs when small blood vessels in the white part of the eye are damaged and burst. This looks a lot worse than it really is and, generally speaking, a subconjunctival hemorrhage will resolve on its own and doesn't require any further workup.
The most common cause of subconjunctival hemorrhage is probably coughing or straining. This may occur while you sleep, as we often cough while asleep without realizing it. Similarly, you may have accidentally poked or scraped your eye while sleeping, which could also cause the hemorrhage. There is not much you can do to speed up the healing of the subconjunctival hemorrhage - your body will slowly on its own absorb the blood back over a period of days to weeks, depending on the size of the hemorrhage.
If your eye is more generally reddened but without a true collection of blood in the white of the eye, then the other possibility here is conjunctivitis, which can be from allergies or from an infection. This tends to be uncomfortable, though - either itchy or painful. In any case, please go see your primary care doctor
who can confirm for you what is going on.