Rest

rest.rb

module REST
  def self.OAuth(client)
    REST::OAuth.new(client)
  end

  class OAuth < Module
    def initialize(client)
      @client = client
      freeze
    end

    private

    def included(descendant)
      super
      descendant.send(:include, Methods)
    end

    def define_methods
      define_cmp_method
      define_hash_method
      define_inspect_method if inspect
    end

    # Define an #cmp? method based on the instance's values identified by #keys
    #
    # @return [undefined]
    #
    # @api private
    def define_cmp_method
      keys = @keys
      define_method(:cmp?) do |comparator, other|
        keys.all? do |key|
          __send__(key).public_send(comparator, other.__send__(key))
        end
      end
      private :cmp?
    end

    # Define a #hash method based on the instance's values identified by #keys
    #
    # @return [undefined]
    #
    # @api private
    def define_hash_method
      keys = @keys
      define_method(:hash) do | |
        keys.map(&method(:send)).push(self.class).hash
      end
    end

    # Define an inspect method that reports the values of the instance's keys
    #
    # @return [undefined]
    #
    # @api private
    def define_inspect_method
      keys = @keys
      define_method(:inspect) do | |
        klass = self.class
        name  = klass.name || klass.inspect
        "#<#{name}#{keys.map { |key| " #{key}=#{__send__(key).inspect}" }.join}>"
      end
    end

    # The comparison methods
    module Methods
      # Compare the object with other object for equality
      #
      # @example
      #   object.eql?(other)  # => true or false
      #
      # @param [Object] other
      #   the other object to compare with
      #
      # @return [Boolean]
      #
      # @api public
      def eql?(other)
        instance_of?(other.class) && cmp?(__method__, other)
      end

      # Compare the object with other object for equivalency
      #
      # @example
      #   object == other  # => true or false
      #
      # @param [Object] other
      #   the other object to compare with
      #
      # @return [Boolean]
      #
      # @api public
      def ==(other)
        other.is_a?(self.class) && cmp?(__method__, other)
      end
    end # module Methods
  end # class Equalizer
end

I've warped your example a little bit but this should be understandable.
I added a few comments on the code to explain the need for certain things

module Rest
  def self.OAuth(client)
    Rest::OAuth.new(client)
  end
  class OAuth < Module
    def initialize(client)
      @client = client
      define_methods # This is needed to define the so-called dynamic methods. You may rename this whatever you like obviously
    end

    private
    
    def included(base)
      super
      base.send :include, Methods # This is what makes all methods in Methods available to the target class as if it was a plain old mixin module.
    end

    def define_methods # here you define all dynamic methods you need. in this case we only have one and thus this is cumbersome, but as you add more and more...
      define_client_method
    end

    def define_client_method
      client = @client # This is very important, you might be tempted to do a @client on the next line.. that would fail as that would be fetched from the target classe's scope, not what you want
      define_method :rest_oauth_client do # I usually use these "funny/long" names. This is to avoid name clashes. My general rule of thumb here is "<module>_<class>_<accessor>" hence rest oauth client.
        client
      end
    end

    module Methods
      def say
        puts rest_oauth_client
      end
    end
  end
end

class Banana
  include Rest::OAuth("apple")

end

b = Banana.new
b.say

PS: I think it would be possible to build a facilitator that would get you rid of all this boilerplate (the define_methods, the base.send :include, Methods, and possibly even the define_XXX_methods...

Sounds like a nice weekend challenge