Rumor detection on social media has become an emerging topic. Traditional deep learning-based methods model rumors based on content, propagation structure, or user behavior, but these approaches are constrained by limited modeling capacity and insufficient training corpora. Recent studies have explored using LLMs for rumor detection through supervised fine-tuning (SFT), but face two issues: 1) unreliable samples sometimes mislead the model learning; 2) the model only learns the most salient input-output mapping and skips in-depth analyses of the rumored content for convenience. To address these issues, we propose an SFT-based LLM rumor detection model with Influence guided Sample selection and Game-based multi-perspective Analysis (ISGA). Specifically, we first introduce the Influence Score (IS) to assess the impact of samples on model predictions and select samples for SFT. We also approximate IS via Taylor expansion to reduce computational complexity. Next, we use LLMs to generate in-depth analyses of news content from multiple perspectives and model their collaborative process for prediction as a cooperative game. Then we utilize the Shapley value to quantify the contribution of each perspective for selecting informative perspective analyses. Experiments show that ISGA excels existing SOTA on three datasets.