Humanoid social robots are increasingly present in daily life, making sustained user engagement a critical factor for their effectiveness and acceptance. While prior work has often examined affective evaluations or anthropomorphic design, less is known about the relative influence of dynamic conversational qualities and perceived robot characteristics in determining a user's intention to re-engage with Large Language Model (LLM)-driven social robots. In this study, 68 participants interacted in open-ended conversations with the Nadine humanoid social robot, completing pre- and post-interaction surveys to assess changes in robot perception, conversational quality, and intention to re-engage. The results showed that verbal interaction significantly improved the robot's perceived characteristics, with statistically significant increases in pleasantness ($p<.0001$) and approachability ($p<.0001$), and a reduction in creepiness ($p<.001$). However, these affective changes were not strong and unique predictors of users' intention to re-engage in a multiple regression model. Instead, participants' perceptions of the interestingness ($β=0.60$, $p<.001$) and naturalness ($β=0.31$, $p=0.015$) of the robot's conversation emerged as the most significant and robust independent predictors of intention to re-engage. Overall, the results highlight that conversational quality, specifically perceived interestingness and naturalness, is the dominant driver of re-engagement, indicating that LLM-driven robot design should prioritize engaging, natural dialogue over affective impression management or anthropomorphic cues.