AWS Elastic Beanstalk change RDS Endpoint

Update

The topic remains complex and the AWS Elastic Beanstalk (EB) documentation could still do a better job to clarify available options. The question has been about how to change an RDS endpoint, which seems to be read in two different ways:

1. One could interpret it about how to attach an existing externally managed RDS endpoint to an existing (not new!) EB environment - this is indeed not possible, rather one would need to resort to handling this scenario from within the app itself as e.g. outlined in section Using an Existing Amazon RDS DB Instance with Python within Using Amazon RDS with Python.
2. Rather, the OP asked about how to do that after the RDS database was deleted or should be replaced with a new RDS database, i.e. the RDS endpoint change is implied in the process of creating a new RDS database for an existing EB environment that already had one - this is indeed possible by means of the DBSnapshotIdentifier Option Value, which denotes The identifier for the DB snapshot to restore from. Once again the EB docs aren't exactly conclusive what this means, however, EB is using AWS CloudFormation under the hood, and the resp. entry for AWS::RDS::DBInstance - DBSnapshotIdentifier provides more details:
   

   By specifying this property, you can create a DB instance from the specified DB snapshot. If the DBSnapshotIdentifier property is an empty string or the AWS::RDS::DBInstance declaration has no DBSnapshotIdentifier property, the database is created as a new database. If the property contains a value (other than empty string), AWS CloudFormation creates a database from the specified snapshot. If a snapshot with the specified name does not exist, the database creation fails and the stack rolls back.

In other words, the typical result of updating any of the General Option Values from namespace aws:rds:dbinstance for an existing EB environment is the creation of a respectively adjusted RDS instance managed by EB, and thus a new RDS endpoint.

• A specific sub scenario is the use of DBSnapshotIdentifier, which yields a new RDS instance managed by EB based on the referenced snapshot and can therefore be used to migrate (rather than attach) an existing externally managed RDS instance, albeit with considerable downtime based on the snapshot size.

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Initial Answer

While unfortunately not specifically addressed within Configuring Databases with AWS Elastic Beanstalk, the AWS Elastic Beanstalk settings for an optional Amazon RDS database are handled via Option Values, see namespace aws:rds:dbinstance within General Options.

While the AWS Management Console hides many of those option values behind its UI, you can specify them explicitly when using the API via other means, both when creating an environment as well as when updating one (which is how you would change any settings of an RDS database instance) - see e.g. parameter --option-settings for update-environment from the the AWS Command Line Interface:

If specified, AWS Elastic Beanstalk updates the configuration set associated with the running environment and sets the specified configuration options to the requested value.

I created a config file under .ebextensions folder that had the following content:

option_settings:
  - namespace: aws:rds:dbinstance
    option_name: DBSnapshotIdentifier
    value: <name-of-snapshot>

Upload and deploy and it will create a new RDS db using this snapshot.