Updated answer

Here's a slightly modified version of your code. No include, extend, append_features or module_function are needed. It wouldn't be hard to add custom_include with the same structure.

UPDATE: Make sure to read @7stud's answer, with a similar structure and very good explanation.

class Class
  def custom_extend(module_name)
    module_name.instance_methods(false).each do |method|
      define_singleton_method(method) do |*args, &block|
        module_name.instance_method(method).bind(self).call(*args, &block)
      end
    end
  end
end

module SimpleModule
  def class_hello
    puts "hello from #{@name}"
  end
end

class SimpleClass
  @name = 'class'
  custom_extend SimpleModule

  def self.test
    class_hello
  end
end

SimpleClass.test
#=> hello from class

Original answer

Usual way

The usual way would be :

module SimpleModule
  def class_hello
    puts "hello from #{@name}"
  end
end

class SimpleClass
  @name = 'StackOverFlow'
  extend SimpleModule

  def self.test
    class_hello
  end
end

SimpleClass.class_hello

but you don't want it. (why?)

Your way

In your code, SimpleClass and SimpleModule are totally independent from one another. It's clear that you get a NameError. You need to somehow pass the name information.

As name parameter :

module SimpleModule
  def self.class_hello(name='')
    puts "hello from #{name}"
  end
end

class SimpleClass
  @@name = 'StackOverFlow'
  def self.test
    SimpleModule.class_hello(@@name)
  end
end

With klass parameter :

module SimpleModule
  def self.class_hello(calling_class=self)
    calling_class.class_eval{
      puts "hello from #{@name}"
    }
  end
end

class SimpleClass
  @name = 'StackOverFlow'
  def self.test
    SimpleModule.class_hello(self)
  end
end

SimpleClass.test

With binding parameter :

module SimpleModule
  def self.class_hello(b)
    puts "hello from #{b.eval('@@name')}"
  end
end

class SimpleClass
  @@name = 'StackOverFlow'
  def self.test
    SimpleModule.class_hello(binding)
  end
end

SimpleClass.test

With my_ruby_extend SimpleModule

It surely can be done with a custom my_ruby_extend. You'd need to show the desired syntax though, and what you already implemented.

This way, you could tell Ruby that SimpleClass and SimpleModule are linked. When a method or a variable isn't found in SimpleModule, it could be sought in SimpleClass.

I just try to re-implement extend as an exercise.

ruby's extend() doesn't work like this:

module SimpleModule
  def self.class_hello_world
    puts 'i am a simple module boss'
  end

  def self.class_hello_name
    puts "hello from #{@@name}"
  end
end

class SimpleClass
  custom_extend  SimpleModule

For example, the following doesn't work:

module Dog
  def self.greet
    puts "hello"
  end
end

class Cat
  extend Dog
end

Cat.greet

--output:--
`<main>': undefined method `greet' for Cat:Class (NoMethodError)

extend() works like this:

module Dog
  def greet
    puts "hello"
  end
end

class Cat
  extend Dog
end

Cat.greet

--output:--
hello

In other words, extend() inserts the module instance methods--not the module methods(e.g. method names preceded by self)--into Cat's singleton class (which is where Cat's class methods live). In ruby, include() and extend() have nothing to do with module methods (again, method names preceded by self). Modules have two uses in ruby:

1. As a namespace, e.g. containing def self.method_name
2. As a mixin, e.g. containing def some_method

include() and extend() deal with #2.

The following solution doesn't work with @@variables, but trying to figure out all the twist and turns that @@variables exhibit in ruby is not worth the effort--just don't use them. Use class instance variables instead, i.e @variables specified outside of any def's:

def my_extend(some_module)
  singleton_class.include some_module
end

module Dog
  def greet
    puts @greeting
  end

  private
  def sayhi
    puts "hi"
  end
end

class Cat
  @greeting = "hello"
  my_extend Dog
end

Cat.greet
#Cat.sayhi  #=>`<main>': private method `sayhi' called for Cat:Class (NoMethodError) 

Cat.class_eval {sayhi}  #Change self to the Cat class so the implicit
                        #self variable that calls sayhi is equal to Cat

--output:--
hello
hi

Now, you just need to implement my_include and substitute it inplace for include. :)

Here's a shot at my_include():

class Class

  def my_include(module_)
    #For public and protected methods:
    module_.instance_methods(include_super=false).each do |meth_name|
      meth = module_.instance_method(meth_name)
      define_method(meth_name) do 
        meth.bind(self).call
      end
    end

    #For private methods:
    module_.private_instance_methods(include_super=false).each do |meth_name|
      meth = module_.instance_method(meth_name)
      define_method(meth_name) do
        meth.bind(self).call
      end
      private :"#{meth_name}"
    end

  end
end

module Dog
  def greet
    puts "hello"
  end

  def go
    puts "run, run run"
  end

  private
  def sayhi
    puts "hi"
  end

end

class Cat
  my_include Dog
end

c = Cat.new
c.greet
c.go
c.sayhi 

--output:--
hello
run, run run
 #=>`<main>': private method `sayhi' called for #<Cat:0x007fc014136f60> (NoMethodError)

With my_extend():

class Class

  def my_include(module_)
    #For public and protected methods:
    module_.instance_methods(include_super=false).each do |meth_name|
      meth = module_.instance_method(meth_name)
      define_method(meth_name) do 
        meth.bind(self).call
      end
    end

    #For private methods:
    module_.private_instance_methods(include_super=false).each do |meth_name|
      meth = module_.instance_method(meth_name)
      define_method(meth_name) do
        meth.bind(self).call
      end
      private :"#{meth_name}"
    end

  end

  def my_extend(module_)
    singleton_class.my_include module_
  end

end


module Dog
  def greet
    puts @greeting
  end

  private
  def sayhi
    puts "hi"
  end
end

class Cat
  @greeting = "hello"
  my_extend Dog
end

Cat.greet
#Cat.sayhi  #=>private method `sayhi' called for Cat:Class (NoMethodError)
Cat.class_eval {sayhi}

--output:--
hello
hi

Simplify question

You can simply your question by modifying your code as follows.

module SimpleModule
  def self.class_hello
    puts "hello from #{@@name}"
  end
end

class SimpleClass
  @@name = 'StackOverFlow'
end

SimpleModule.class_hello
  # NameError: uninitialized class variable @@name in SimpleModule

Which class?

Clearly, the module's module method class_hello must be informed of the class whose class variable &&name is desired. We therefore must give that class as an argument of class_hello. We then use [Module#class_variable_get](https://ruby-doc.org/core-2.2.1/Module.html#method-i-class_variable_get ) to extract the class variable's value.

module SimpleModule
  def self.class_hello(klass)
    puts "hello from #{ klass.class_variable_get(:@@name) }"
  end
end

SimpleModule.class_hello(SimpleClass)
  # hello from StackOverFlow