You can create a Ruby function that allows the user to modify the backup retention period for an Amazon RDS database instance using the AWS SDK. The following example demonstrates how to achieve this:

require 'aws-sdk'

def modify_rds_backup_retention
  rds = Aws::RDS::Client.new(region: 'us-east-1')  # Replace 'us-east-1' with your desired region

  # Get the RDS instance identifier from the user
  print 'Enter the RDS instance identifier: '
  instance_identifier = gets.chomp

  # Get the desired backup retention period from the user (in days)
  print 'Enter the new backup retention period (in days): '
  retention_period = gets.chomp.to_i

  # Modify the backup retention period for the specified RDS instance
  rds.modify_db_instance({
    db_instance_identifier: instance_identifier,
    backup_retention_period: retention_period
  })

  puts "Backup retention period modified for #{instance_identifier} to #{retention_period} days."
rescue StandardError => e
  puts "Error modifying backup retention period: #{e.message}"
end

modify_rds_backup_retention

In this Ruby function:
* We require the `aws-sdk`' gem to use the AWS SDK for Ruby.
* A client for Amazon RDS is created using `Aws::RDS::Client.new`. You should replace 'us-east-1' with your desired AWS region.
* The user is prompted to enter the RDS instance identifier and the desired backup retention period (in days).
* The `modify_db_instance` method is called to modify the backup retention period for the specified RDS instance.
* After the modification is successful, the program provides a confirmation message.

Make sure to configure your AWS credentials using the AWS CLI or environment variables to allow the SDK to authenticate with your AWS account.