The easy part of your question is the eligibility.
"Redshirt," which is not an NCAA term, refers to a year that does not count as one of an athlete's four permitted seasons of competition. The medical hardship mentioned by strainedoblique is a provision that permits athletes who have season-ending injuries in the first half the season not to count that season as a season of competition if they haven't played in 30% of the team's games for the whole season.
If your cousin's son doesn't play at all next year, it won't count was one of his four seasons of play. If he comes back and plays at all, it would count as one of his seasons of play.
The hard part is what will happens to his baseball opportunity if he takes a leave of absence from school.
Coaches look forward, not backward. Except in very rare instances, such as encouraging a player to go to a JUCO to get more playing time, coaches assume players who leave school aren't coming back. It's extremely unlikely any coach would set aside scholarship money for an athlete who left the school and won't be ready to play in the spring if he does come back. That money will go to someone who will be there.
Maintaining relationships with coaches and teammates during rehab is extremely important. If the coach doesn't see day-in, day-out commitment to rehab and the team, your cousin's son will be out of sight and out of mind.
Unless there are compelling circumstances requiring it, taking a semester off is a very bad idea if your cousin's son wants to play for his team again. From the coach's point of view, it signals lack of commitment to school and lack of commitment to the team.