I don't think the spectator has a case against either player.
I haven't looked up the suit to identify the alleged causes of action, but based on the facts I know about I'm willing to speculate that there are three possible counts. Namely: 1) tortious battery; 2) negligence (also a tort); and 3) intentional infliction of emotional distress (father on behalf of son). I don't think any of them is a winner.
In most jurisdictions, tortious battery requires the plaintiff to demonstrate that the defendant's intentional act caused a harmful or offensive touching to the plaintiff's person, or some object attached to the plaintiff's person. Here, while the batter intentionally swung the bat, he did not do so with the intent of hitting the spectator. Thus, the player lacked the requisite intent required by the tort of battery. This is contrary to cases of true intentional battery, such as the Manning v. Grimsley case (Orioles pitcher intentionally threw ball at a Red Sox fan who was heckling him) available at
http://ftp.resource.org/courts.gov/c/F2/643/643.F2d.20.80-1145.htm .
A negligence action requires the plaintiff to prove 5 elements, namely duty, breach of duty, actual causation, proximate causation, and damages. Duty relates to the standard of care the defendant owed to the plaintiff, if any. While an elevated duty of care might be imposed on the MLB to ensure spectator safety (because specators are business invitees to an event hosted by the MLB), I am not aware of any similar elevated standard that applies to players. Thus, the player in this case will likely be held to only owe the spectator a duty of ordinary care (i.e., to act in the same way that a reasonably prudent person would act under the same circumstances).*
Breach goes to whether the defendant failed to exercise the required duty of care. By using a bat that is known to have a propensity to break, it is arguable that the player did not exercise the required standard of care. But given the prevalence of the use of such bats in the industry, it is a very close case for the judge or jury to decide. My opinion is that the player wins on the elements of duty and breach, and thus wins the whole case. The player also may be able to assert that the defendant "assumed the risk" of being hit with a bat when he entered the park and remained in his seat, despite MLB warnings that bats may leave the playing area.
Causation and proximate cause relate to whether the defendant actually caused the plaintiff's injury, and whether the plaintiff was foreseeable. Here, there is little question that the player's bat caused the injury. But for the player swinging the bat and contacting the ball, the bat would not have shattered and injured the spectator. While it could not be predicted that any one specific spectator would be hit, the fact that a broken bat might fly into the stands and injure "a" spectator was predicatable (as evidenced by the numerous pictures in this thread). Thus, causation and proximate cause are likely met.
Damages - not an issue in this case. The spectator was clearly injured.
As to intentional inflication of emotional distress, that tort usually requires an intentional act by the defendant (which in some cases must arise to extreme or outrageous conduct) that results in damages to the plaintiff in the form of extreme emotional distress. In bystander cases, such as this one, the plaintiff is often required to show that the defendant knew the plaintiff was there, and went through with the act causing the extreme emotional distress anyway. Where the emotional distress arises by conduct to a third party, the plaintiff and the third party are frequently required to have a close relationship (met in this case by the fact that the plaintiff is the son of the person who was injured).
Here, it is doubtful that the plaintiff can prove that the player's acts were "extreme" or "outrageous" or even that the player was aware of the plaintiff's presence. Given these substantial obstacles, I do not see the plaintiff (son's) case for emotional distress going anywhere.
What I am surprised about is that the plaintiff has not sued the manufacturer of the bat for products liability. From the limited knowledge I have on the subject matter, it would seem that the plaintiff has a pretty good strict products liability case based on a design defect in the bat. Not to mention that the manufacturer likely has much deeper pockets than either of the players involved.
Final sidenote. The suit against the player who lent the batter the bat is likely based on a "negligent entrustment" theory. That theory usually requires the plaintiff to show that the defendant had at least some knowledge (scienter) of the (usually hidden) dangerous propensities of the object being lent, or the party to whom the object is being lent (e.g., parent A lends matches to a small boy B, knowing that B is a pyromaniac. B burns down neighbor C's hous with the matches. C can likely sue A for negligent entrustment).
* Note, if anyone wonder's why the NHL started putting nets up around hockey rinks in the mid 1990's, this is why.
**Note 2: none of the above is legal advice or intended as legal advice. Any reliance on it is at the reader's own risk.