The ability of accurate depth prediction by a convolutional neural network (CNN) is a major challenge for its wide use in practical visual simultaneous localization and mapping (SLAM) applications, such as enhanced camera tracking and dense mapping. This paper is set out to answer the following question: Can we tune a depth prediction CNN with the help of a visual SLAM algorithm even if the CNN is not trained for the current operating environment in order to benefit the SLAM performance? To this end, we propose a novel online adaptation framework consisting of two complementary processes: a SLAM algorithm that is used to generate keyframes to fine-tune the depth prediction and another algorithm that uses the online adapted depth to improve map quality. Once the potential noisy map points are removed, we perform global photometric bundle adjustment (BA) to improve the overall SLAM performance. Experimental results on both benchmark datasets and a real robot in our own experimental environments show that our proposed method improves the overall SLAM accuracy. While regularization has been shown to be effective in multi-task classification problems, we present experimental results and an ablation study to show the effectiveness of regularization in preventing catastrophic forgetting in the online adaptation of depth prediction, a single-task regression problem. In addition, we compare our online adaptation framework against the state-of-the-art pre-trained depth prediction CNNs to show that our online adapted depth prediction CNN outperforms the depth prediction CNNs that have been trained on a large collection of datasets.