Final answer:
The cyclists' dialogue about distinguishing their names and the subsequent discussion on their parents and ages suggest a familiarity that raises the question of a possible familial connection.
Step-by-step explanation:
The detail in the cyclists' dialogue that refines the idea that they might be related is when they discuss needing to differentiate their names, "You shall be Ted, and I Ned" followed by the exposition about their ages and parents. The natural familiarity with which they entertain and set upon the idea of having similar names and the details about their parents and ages suggest a bond that could imply a familial relation. The moment that strikes curiosity is when one cyclist mentions, "we can't be complicated up as long--lost brothers--can we?" The frowning, sharp response from Ted to this suggestion further piques the interest in their relationship.
The detail in the cyclists' dialogue that refines the idea that they might be related is when one of them suggests differentiating their names. He proposes that they use the names Ted and Ned, which are both dentals but philologically uninterchangeable. This shows that they have a familiarity with each other and have likely spent time together, indicating a possible familial relationship.