What method allows sharing of a custom object record with a specific site user?

Manual sharing is the correct choice for sharing a custom object record with a specific site user. This method allows users with appropriate permissions to share individual records with other users, regardless of the sharing settings that are in place for the object.

When a record is shared manually, the owner of the record or a user with the necessary permission can add the specific user to the sharing list for that record. This flexibility is crucial in scenarios where you need to grant access to a few users without changing the default sharing rules for the entire object.

In contrast, the other choices involve different mechanisms of sharing. Direct sharing usually refers to sharing at a more automated or organization-wide level, which may not provide the granularity needed for individual records. Role accessibility pertains to sharing records based on user roles, which can affect a broader scope of users rather than targeting one specific individual. Automatic sharing happens automatically based on predefined rules and may not allow for the selective sharing of records with specific users. Hence, manual sharing is the most effective method to achieve the desired outcome in this scenario.