Cardiac T1 mapping by magnetic resonance imaging (MRI) is an important clinical tool for the diagnosis and treatment of cardiovascular diseases. In practice, involuntary cardiac and respiratory motion often results in reduced accuracy and precision in T1 estimation. Motion correction is an essential preprocessing step, however, with intensive contrast changes among baseline images, both optimization-based and deep-learning (DL)-based registration methods still struggle to estimate structural similarity between images, especially when image contrast is poor and displacement is large. In this work, we propose a novel registration metric that is highly insensitive to large contrast changes, based on modified modality independent neighborhood descriptor (mo-MIND). To accommodate severe motions, we further propose pre-deformation as an augmentation strategy at the training stage. We combine the proposed mo-MIND-based metric and the augmentation strategy in a U-Net architecture to tackle the challenges of motion correction for cardiac T1 mapping. Experimental results and ablation studies demonstrated that our method achieved improved registration performance compared to several established baselines, leading to significantly reduced T1 mapping error and improved landmark stability.