Change a string into a function/def activator
There are quite a lot of things you should do differently.

Let's start with your function random_number. You should not overwrite the string module (even though it is only locally), which you imported. You should also use list comprehensions to build the string. To pretty print the number as a phone number, use str.format. Use the fact that random.randrange takes an implicit 0 as its first argument (just like range). And finally use tuple unpacking, instead of manually indexing the list.

def random_number():
    phone_number = [random.randrange(10) for _ in range(10)]
    return "({}{}{}){}{}{}-{}{}{}".format(*phone_number)




Next, your random_keyword function. You can vastly simplify the building of random_letter by using a list comprehension again, together with str.join, to avoid costly string addition.

Your while True loop also serves no purpose, since you break it in either case. 

Also, what is the point of storing something in a dictionary (with \$\mathcal{O}(1)\$ lookup time if a key is in it) and then checking to see if something is in the values, which is a list and takes therefore \$\mathcal{O}(n)\$? At this point your dictionary is just a simple list. You can access its members with numbers from 0 to len(accepted_keywords). Instead have a list of your functions and have a dictionary with their names as keys (which you can build using a dictionary comprehension).

functions = [nothing, abc, deft, ghi, jkl, mno, pqrs, tuv, wxyz]
accepted_keywords = {f.__name__: f for f in functions}


def random_keyword():
    keyword = ''.join(random.choice(string.ascii_lowercase)
                      for _ in range(random.choice([3, 4])))
    print("keyword: {}".format(keyword))
    print("*Approved*\n" if keyword in accepted_keywords else "Denied\n")




Finally, Python has an official style-guide, PEP8. It recommends using ALL_CAPS for global constants (like accepted_keywords).

Python also has a docstring convention (written down in PEP257), which in its simplest form looks something like the following:

def random_number():
    """creates a random phone number"""
    ...


At this point you should also notice that that function name is not very descriptive, maybe something like random_phone_number instead?