The argument is advanced that Jesus' Apostles were telling the truth of what they had witnessed because they died martyrs' deaths and would not have done so for a lie.
This is a valid point, but I think what it is meant to prove is sometimes misunderstood so I just want to clarify the purpose of the argument. It's been pointed out many times that people of other religions die for their faith so the disciples' deaths don't prove anything. Other religions have martyrs, too, but that doesn't mean those religions are true. And many Christians aren't willing to die for their faith. So this argument doesn't prove anything. But this response misunderstands the point of the argument.
This is not a proof for the truth of Christianity. It's an evidence for the Apostles' sincerity in what they were testifying about Jesus. It is a rebuttal to the charge that the Apostles lied and made up what they'd seen. People don't die for a lie, but they might be willing to die for what they believe firmly to be true.
This rebuttal doesn't even prove that the Apostles' testimony was true - only that they believed it was true and didn't lie. So the fact that people of other religions also die for their faith doesn't undermine the significance of the Apostles' martyrdom for their testimony - these people, too, must have strong convictions. The fact that many Christians might not be willing to die for our faith doesn't undermine this argument because it is only relevant for those eye witnesses who passed on what they saw and heard at the cost of their lives. The Apostles believed what they told us and did not lie.
It's not a proof for the truthfulness of Christianity, but a rebuttal to the charge they lied. And so it is one of the elements in evaluating the historical evidence for Jesus' death and resurrection.