Emotion recognition based on neural signals is a promising technique for the detection of patients' emotions for enhancing healthcare. However, emotion-related neural signals, such as from functional near infrared spectroscopy (fNIRS), can be affected by various psychophysiological and environmental factors. There is a paucity of literature regarding data instability and classification instability in fNIRS-based emotion recognition systems, phenomenon which may lead to user dissatisfaction and abandonment. We collected data in an fNIRS-based 2-class emotion recognition test-retest experiment (3 week interval) with visual stimuli emotion induction to examine data instability and its impact on classification accuracy. We found a 22.2% average deterioration of emotion classification accuracy between the two sessions, suggesting that classification instability is a serious problem. We found that the changes in the distributions of the selected neural signal features, as evaluated by Kullback-Leibler (KL) divergence, were a likely cause of the accuracy decline. We analyzed the data instability and our results showed that instability of spatial activation patterns and instability of the hemodynamic response in the most activated region are correlated with accuracy decline. Finally, we propose a method for mitigating classification instability in fNIRS-based emotion recognition based on feature selection for stable features, the first such method to our knowledge. This new feature selection criterion considers not only the separability of features (evaluated by Fisher Score) but also their stability over time (evaluated by KL divergence between feature distributions at different time points). Testing showed that this method led to an approximately 5% improvement in cross-session generalization accuracy.