The execution of a single crossbowman probably had nothing to do with the weapon, and far more to do with the politics of fealty and capital punishment. Although the chronicles (though for the Siege of Rochester we're mainly talking about the Barnwell/Crowland Chronicle) don't name the guy John had executed, they do give a telltale bit of information. The chronicle's wording is a little weird (the Latin is Unum solum suspendi iussit, arcubalistarium quem, ut fertur, a puero nutrierat, cum tamen putaretur pre ire magnitudine quod omnes simul morti amarissime mox traditurus esset., if anyone's interested), but the implication is clear: this crossbowman was killed because had been in the king's household and betrayed him to join the rebelling barons. John and this crossbowman knew each other, and his execution was personal.

Whether someone deserved execution by the norms of medieval war could be a delicate matter. Ordinarily, those who resisted a siege could only expect mercy in the event of a negotiated surrender, which if they were resisting was a bit unlikely. The defenders during the Siege of Rochester were certainly not in the negotiating mood until the very end, by which time the opportunity to negotiate surrender had usually passed. The general rule was that all bets were off when the walls were breached, but the defenders kept fighting after that point. John's forces had to commit considerable forces to the siege; several large siege engines that hammered the walls and towers day and night, and they undertook an ambitious sapping operation to bring down the fortifications. By that point, most commanders would have wanted to make an example of the defenders. In addition to this, the punishment for treachery was death, no ifs or buts. In John's eyes, the rebels were committing treason against the crown so they ought to be executed. As was often the case when John did something controversial, he could justify it by saying it was legal and therefore morally acceptable within the laws and customs of England.

But this was not an ordinary war, or an ordinary rebellion. The defenders of the castle were rebelling against John because they believed he had broken the principle of fidelitas, which is most obviously translated as fidelity but sometimes as fealty. It's the glue that held medieval politics together, and under its rules a rebellion could be entirely legitimate as a fight against tyranny. John had done many things that were clearly not ok even though they were technically legal. He had tried to confiscate the land of loyal vassals. He had trapped some of his noblemen in debt. He ignored long established conventions and charters regarding the ownership of certain castles and cities (including Rochester - it had traditionally been held by the Archbishop of Canterbury but John demanded possession of it). By 1215 the barons of England had enough, and many of them felt that John had broken the bonds of fealty by acting in bad faith toward them, and had done so to the point that he needed to be violently taught a lesson in proper treatment of one's vassals. At that point a rebellion could be seen as legitimate, and although John's own legalistic view meant he did not think the rebellion was legit, many of the nobles who stuck by his side thought the rebels had a point. Many of them had the same issues with John as the rebels, they just didn't think it was something to mobilise their armies over. Some of them, notably the great Earl William Marshal of Pembroke, used supporting John as leverage to get him to return some of the castles he'd arbitrarily confiscated. In the eyes of most English noblemen, the rebels were not traitors and did not deserve death because the conflict was more of a political dispute than a proper war. And with that came a level of restraint that wasn't normal.

So when the garrison of Rochester Castle finally gave in, John wanted to kill them all as traitors. But one of his most trusted advisors, Savari de Mauléon, interceded and talked John down from doing it. Although we don't get the full discussion, we do get that Savari's main argument was one of escalation. If they killed the garrison, it would signal to the rebel barons that John was now operating according to the norms of war with all the brutality that entailed, and that the rebels could retaliate. The First Baron's War had so far been pretty tame; there hadn't been the ravaging of the land that was normal at the time, and non-combatants and wounded had generally been allowed to leave besieged places (including Rochester). They weren't tearing up England and killing en masse like they had in the previous civil war back in the 1140s. Savari was concerned that John executing the garrison would change that, and John was persuaded.

Which then raised the issue of the lone crossbowman. He was in the king's own household, on his payroll, and here he was shooting crossbow bolts at his own comrades in John's personal forces. He was unambiguously a traitor to his lord, so he had to hang even if everyone else could merely be imprisoned.