"best friends"

It’s written as two words with a space between them, but it’s considered to be a compound noun. A compound noun is a noun that is made up of two or more words, often either two nouns or an adjective plus noun. Sometimes the words in a compound noun are separate, sometimes hyphenated, and sometimes physically joined together.

So, in this case, 'best friends' is treated as a single word, e.g., They have become acquaintances, enemies, partners, best friends, etc.

If one wishes to use "the", then the expression would become "the best of friends".

Hi Gabriel,

I'm going to take a shot at this and see if I can help! Although I'm a mother-tongue English speaker, I've never really asked myself (or anyone else) this question. But it helps to think of it like we would when naming people as members of a family (e.g., "They are sisters" or "They are parents") or as any kind of identifying role (e.g., "They are athletes"). In these cases, the nouns that follow the verbs are subject complements. In the single form, a subject complement requires an article (I am a cook or She is a student) but the plural forms do not (They are students). Think of it as assigning a sort of job title to the plural subject. We are not speaking of a specific cook or specific student but rather just giving another way to name/identify them. In plural forms, subject complements do not require articles unless we are referring to an exact or specific noun. It would require an article if we were to say, for example, "They are the best friends who went to the concert together." In this case we are specifying them further.

Hope this helps!

Sarah 

having said all that "they have become best friends" is fine too. but "they have become THE best OF friends" expresses in English how extra valued their friendship is between them. 

it is another way of expressing "they have become VERY GOOD friends" 

Your student is almost correct.

"They have become the best OF friends" = the correct way to say/express this sentiment.

"They are the best of friends"

"They became the best of friends" 

you will be using the second dictionary definition of  the word "of" a scale not a relationship as "in made from/of" "born of" 

expressing the relationship between a scale or measure and a value.

their friendship has become good, true, reliable, meaningful, valued =  a measurable scale

He is a friend of mine = First definition the word "of" in a dictionary = a relationship "a friend of mine"

This is a good example of when the important English word "OF" is used, this is why you will frequently see native English speakers adding in the word "of" into sentences when correcting. Even though the sentence can be read and understood without it.

You should congratulate your student and buy him or her or (binary gender they) a present.

As an afterthought...

If we considered the term "best friend" to be composed of the noun friends and the superlative adjective form of good, i.e., best, then the superlative adjective would require the definite article "the". However, if we consider the term "best friends" to be a compound noun, then there is no superlative adjective and thus no need for the "the".