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

Here's the explanation for the overlooked word:

Step 1: 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."

Step 2: Identify the rumor that spread. Verse 23 states, "Because of this, the rumor spread among the believers that this disciple would not die."

Step 3: Compare Jesus' actual words with the rumor. 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 $\boxed{The one overlooked word that started the rumor in John 21:21-23 was "If". Jesus' statement was conditional ("If I want him to remain alive until I return"), but the rumor misinterpreted it as a definitive declaration that the disciple would not die.}$