In some situations, a common noun in a specific scenario is treated as a proper noun because it refers to a specific entity that satisfies the common noun.

Is there a special term for this phenomenon?

Examples:

"Go ask his father", said the teacher. vs "Go ask Father", said the mother.

and

"Most city halls have them", she replied. vs "City Hall has them", he stated.

In conventional resources on common and proper nouns, they are treated as two separate subcategories of noun.

Some sources do consider converting one kind of noun to another. Here is ThoughtCo author Richard Nordquist on going from a common to proper noun. The only applicable verb in the article is capitalize, referring to the change in form that signals a proper noun:

Many great authors have used the idea of capitalizing common nouns and making them proper to characterize specific inanimate objects or take a concept like "Great Places" and make them into a physical place in a fictional world.

However, in more academic contexts linguists have referred to the creation of proper nouns (single words) or proper names (noun phrases) as naming. Adrienne Lehrer in "Names and Naming: Why We Need Fields and Frames" describes the conventions of naming:

A common basic distinction in noun subclasses is between proper and common nouns. Often little more is said about the difference, as in contemporary generative grammar, where the difference is characterized by the feature +/- Proper. However, if we look at a wide range of names and at the processes for naming, we discover that the difference between common and proper nouns is anything but clear-cut; and moreover, the vocabulary is not neatly divided.

Another author, Willy Van Langendonck, describes this process as naming when summarizing a theory of proper name formation in Theory and Typology of Proper Names:

Thus, the difference between proper nouns and common nouns is basically a difference of dénomination (naming), i.e. proper names should be defined in terms of the (instructional) meaning X called Y, but not common nouns.

So naming forms a proper name or noun from (usually) a common noun or (occasionally) other parts of speech.

The reverse process is appellatization, described in The Dictionary of Historical and Comparative Linguistics as

The conversion of a proper name into a common noun

and cites aspirin and Kleenex as examples. Its verb form (appellativize) is turgid and technical.

It is following capitalization rules for nouns, and does not have a specific term.

The role of noun is not tied to the word, but to what it refers to. Words like mother and father are common nouns when the word "refers to people or things in general. When they refer to a particular person, they are a proper noun.

In your example, the capitalized use "Father" is being used as a proper noun as it identifies a particular being.