Well, first, you should know that bulging eyes are well known as Proptosis, or exophthalmos. And it can occur in adults or babies. For many people, it is the bulging out of one or both bilateral of the eyes. And we can say that there should be no visible white between the top of the iris, which is the colored part of the eye, and the upper eyelid. In some situation when you can see white in this area, it is usually a sign of bulging eyes. Of course, it will just like your baby gets. For the causes, glaucoma, Graves Disease, Hemangioma, Histiocytosis can be possible. And if your baby has it, I will suggest you take your baby to see the eye doctor as soon as possible, for it is considered a very serious problem.
You should better take your three-month-old son to go to the hospital to have a full check on the eyes. The bulging eyes may be caused by the eyes infection because of the invisible bacterium which you need to take great notice of. The sooner your son accepts the treatment, the better his situation of the eyes is.
Does he has any other symptoms like eye pain, eye redness, or other discomfort? If so, he may be crying all the time even when he is not hungry. Observe his eyes for more information. If you find he feels uncomfortable, you need to take him to hospital as soon as possible. But I have to say that it is common for babies to have bulging eyes as they have a very small face. When they grow older, their eyes will seem smaller.
also, the causes of bulging eyes include hyperthyroidism, glaucoma, leukemia, and so on. As hyperthyroidism is hereditary, you must think if any family member had ever had this disease.