Sure. It’s a diplomatic wrangle and there are many benefits, but not all of them are benefits we have deliberately sought out. They are kind of side-effects of the Dalai Lama’s stay in India.
It is not uncommon for competing nations to intrude on each other’s authority in many small ways. For example, every time China says that the Indian state of Arunachal Pradesh is part of its territory, it infringes on India’s authority a little.
By allowing political asylum to the Dalai Lama, India undercuts China’s authority (or claim thereof) a little. But that’s not the only reason.
The Dalai Lama is a refugee in India. This means that he is here to avoid arrest by Chinese authorities. His existence gives hope to the Tibetan cause and irks China, who refuses to acknowledge Tibet as an independent political unit. His remaining in India makes India the seat of the pro-Tibet movement and brings us global goodwill from anyone who sees China as the bully in this context. This goodwill also translates into people seeing China as the bully on other matters like its claim on Arunachal Pradesh.
What the Dalai Lama has given India is a moral advantage against China.