It's certainly done a lot in every-day speech. But is it acceptable to say:
I'm happy today, me.
or, to take it to another level:
He was well behaved, my son, this morning.
And, if so, is there a name for this construction?
It's seldom unacceptable to say or write anything that clarifies meaning. The features you describe are known as tails, and are normally found only in speech. ‘They occur at the end of clauses, normally echoing an antecedent pronoun and help to reinforce what we are saying’ (Carter, R. 'Grammar and Spoken English’ in Applying English Grammar). The tail in the first of your examples, me, echoes the antecedent pronoun I, while in the second, my son, it echoes he (although the way you have presented the example doesn’t place it at the end, where it might perhaps more naturally occur).
I'm not sure why your question asks specifically about nouns. This construct could be used, not with just nouns or single words, but for clauses and other parts of speech as well.
One punctuation guide1 says:
The construct you've described in your question seems to be doing that same thing, only at the end of the sentence.
But it's not just an extra noun that can be tacked on at the end:
I'd be careful not to overuse such a construct, but it's well within the realm of “acceptable,” I think.
If a sentence is at all ambiguous, it is always a good idea to add words to clarify the meaning. Even if a sentence is not technically ambiguous, in the sense that if someone reads it carefully they should understand the intended meaning, it may be desirable to add words to call atttention to a point where someone might make an incorrect assumption.
In spoken English, we sometimes make a statement and then realize that it is unclear, and so add words to the end to clarify. But in written English, the words should be added in the proper place in the sentence to make it grammatically correct. After all, when you are speaking, once the words are out of your mouth it's too late to go back and change them. But in writing we can edit before we publish.
So yes, I might SAY, "He was well-behaved this morning, my son." (Well, I doubt I would ever have said that about my son, but that's another story. :-) But if I was writing it and I thought "he" might be ambiguous, I would probably write, "My son was well-behaved this morning." If I thought that was still insufficient, perhaps that someone reading it quickly might gloss over the "my" and think I was talking about my neighbor's son or some such, I might add yet more words, "My son, the child that I myself fathered 12 years ago, ..." etc.