Read John 21:21-23 to understand the context.

In this passage, Peter asks Jesus about the fate of the "disciple whom Jesus loved" (traditionally identified as John). Jesus responds, "If I want him to remain alive until I return, what is that to you? You must follow me."

Verse 23 states, "Because of this, the rumor spread among the believers that this disciple would not die."

Jesus' statement was conditional: "IF I want him to remain alive..." The rumor, however, took this as a definitive statement that the disciple would not die, omitting the crucial conditional aspect.

The one overlooked word that started the rumor was:
If

The final answer is $\boxed{\text{The one overlooked word that started the rumor in John 21:21-23 was "If". Jesus' statement was conditional ("If I want him to remain alive until I return"), but the rumor misinterpreted it as a definitive declaration that the disciple would not die.}}$