Is standing by the grave of the deceased and supplicating for them better than supplicating for them anywhere else, and what is the best supplication for the deceased?

There is no difference between supplicating for the deceased at their grave or from afar, except that supplication at the grave adds the virtue of visiting graves, which benefits both the visitor and the visited. The best supplication for the deceased when visiting the grave is that which was narrated from the Prophet, peace and blessings be upon him, such as the supplication: "O Allah, forgive him, have mercy on him, grant him well-being, and pardon him..." This supplication may be recited at any time and is preferable to a supplication that is not authentically transmitted, though it is permissible to supplicate for the deceased with any supplication that seeks forgiveness and mercy.