There is no particular significance or essence of weekly puja for the dead. It is performed by some people on the particular day on which the person died. This is not followed by majority of Hindus.
People who were unable to perform their duties well when the dead person was alive or who have not yet severed the attachment perform weekly puja.
The best option is to do monthly puja on Amavasya or the annual shradh during pitru paksha.