The above answers miss the point and make things too complicated. Both of your phrases are incorrect.
"One of the my favorite song" is incorrect, because what you have is a partitive construction. In other words, you have a part to a whole. Therefore, the whole has to be plural.
Consider this example. You cannot say "one of the man." The reason for this is owing to the fact that you need a plural. "One of the men," would, therefore, be correct.
"One of my favorites song" is also incorrect. Adjectives do not have number. In other words, even if adjectives match singular and plural nouns, they do not change their forms. For example, you can say: "nice songs" or "nice song."
In light of this, the correct phrase should read: "one of my favorite songs."
A phrase is a fragment, or part, of a sentence and is intended to express one single idea. It contains a minimum of two words and is used in language to signify a concept or a more in-depth explanation than can be expressed with just one word.
Phrases and clauses are often confused but a phrase, unlike a clause, does not contain a verb. A phrase cannot stand alone and is supported by the remainder of a sentence. There are different classifications, such as noun phrases, adjectival phrases and adverbial phrases, depending on the idea being relayed. They explain, qualify or modify nouns and verbs in the same way as adjectives and adverbs do, adding detail to the sentence.
An adjectival phrase will give the reader more information about the subject (the noun); for example, "The dog with the shiny coat came bounding towards me." In this sentence the reader learns more about the dog which is the subject (the noun) of this particular sentence.
An adverbial phrase adds detail to the verb. There are three main types, being adverbial phrase of time, place and manner; although there are a few other types. An example of an adverbial phrase of manner is, "The dog entered the yard aggressively and angrily." Note how the phrase tells the reader HOW the dog enters the yard. An adverbial phrase of time gives an indication of WHEN something may have taken place such as in the following example: "He took the dog to the vet the following day." An adverbial phrase of place tells the reader WHERE: "The dog, in the kennel, is sheltering from the rain.
Noun phrases, much like the one in this question, concern the noun in the sentence. They answer the listener's or the reader's assumed questions concerning the noun. "One of my fav songs," or "One of my favorite songs," tells the reader that the song to which the person is referring happens to be a favorite song. The writer or speaker could make reference to a favorite song and the sentence could be, for example, "One of my fav songs is All About That Bass by Meghan Trainor." The speaker is perhaps answering the question, "What is your favorite song?" or "What are your favorite songs?"
The problem of the question that I do not see any of the phrases written correctly. The correct phrase would have been the option that would have shown the words:
One of my favorite songs.
The word "favorite" does not need to be written in plural because it is an adjective that is placed before the noun. However, if the sentence had suppressed the word "songs", you could say:
The song is one of my favorites
Therefore, as long as there is a noun included in the sentence, focus on the noun to convert it to singular or plural. If the adjective is by itself, then you work on the adjective and apply the same rule.