What is the role or the meaning of “less” and “of” in “it's less of a mouthful than 'Henrietta'”?
In “I've always called myself 'Henny' because it's less of a mouthful than 'Henrietta'.”, I am not clear about the role or meaning “of” and “less”.
Let me give some examples of my guess at roleand meaning.
“Less” is a pronoun with “of” (used after words or phrases expressing amount, number) in “a third of all people”, “none of them”, “a kilo of apples” etc.
Less is an adjective with “of” (about; relating to) in “Speaking of Elizabeth, here she is”, “Of her childhood we know very little.” etc.
In "it's less of a mouthful than 'Henrietta',” the word less functions as a determiner in the noun phrase "less of a mouthful." Oxford Dictionary Online defines a determiner as "a word such as the, some, my, etc. that comes before a noun to show how the noun is being used." You can see that less is similar in nature to the example some (note the etc. in Oxford's definition indicating an expanded list of possible determiners). In addition, less in the comparative less than cannot be analyzed in isolation because, in this comparative context, it is part of a formula for comparison, that being less than. Oxford gives patterns for it's use as (1) less of a something than something and (2) in less than something. Further, in this usage, than functions as a preposition (Oxford).
The comparative phrase "less of a mouthful than" fits the Oxford pattern less of something than something. The role of less is to describe the type of comparison in the pattern less of something to more or something else. Its meaning is that of a determiner in a noun phrase; it shows how the noun is being used. In this case, the noun mouthful is being used as the point of (or the origin of) a comparison to another thing. The comparison of the noun to another thing gives clarity on how to perceive, understand, or think about the other thing. In this case, the noun mouthful, in relation to Henny, gives clarity to how to think about Henrietta: it is harder to say; it is too long to say; it is too formal to say; etc.
In "it's less of a mouthful than 'Henrietta',” the word of functions as a preposition in the noun phrase "less of a mouthful." Cambridge Dictionary Online defines of exclusively as a preposition and gives more than a dozen meanings for its use. These include its meanings indicating possession, amount, position, made of, judgement, relating to, done by, felt by, comparing, and many more.
The meaning that pertains to of in "it's less of a mouthful than 'Henrietta'” is that of amount. It is comparing the figurative amount of Henny--remember, we are working with the figurative idiom "less of a mouthful" since you cannot have a literal mouthful of Henny, either the name or the person--to the figurative amount of Henrietta--nor can you have a literal mouthful of Henrietta, name or person. So the role of plays in "less of a mouthful" is that of a connecting preposition. Longman, Cambridge, and Oxford Dictionaries define a preposition as a word before a noun (or a pronoun or a gerund) that shows its relationship or connection to another word. These relationships and connections pertain to time, location, direction, amount, place, position, and method.
Regarding your suggestions as to the syntactical analysis of less and of, first, although Cambridge Dictionary defines less as having a pronoun function, here less is clearly used as a determiner in a noun phrase ("less of a mouthful" noun phrase: determiner + preposition + article + head noun). So your suggestion that less is a pronoun is not correct. None of your examples are syntactically like the noun phrase "less of a mouthful." Second, you have seen that Cambridge Dictionary defines of only, solely, as a preposition. Oxford and Longman agree. So your suggestion that of is an adjective in "less of a mouthful" is not correct. Each of your examples is syntactically very different from "less of a mouthful."