override.rb

class Foo
  def my_method
    puts 'original'
  end
end
module Bar
  def my_method
    puts 'overridden'
  end
end

Foo.send(:include, Bar)

Foo.new.my_method
# original

Would refinements work for your use case?

class Foo
  def my_method
    puts 'original'
  end
end
module Bar
  refine Foo do
    def my_method
      puts 'overridden'
    end
  end
end

using Bar

Foo.new.my_method
# overridden