I have not heard of the term "lonely note" before, but what you're describing seems very similar to a practice sometimes done by our schola director, who has a preference for a somewhat semiological interpretation of the standard Solesmes notation as found in the Liber Usualis.

The real issue with these questions that you're asking is that there are a whole bunch of different theories and schools of thought on the rhythm of Gregorian chant, and most of the people from these different groups would give you different answers.

- If you are following a "classic" interpretation of the Liber Usualis, I believe punctum which have no rhythmic signs are not held longer.

- If you are following the "pure" Vatican Edition (issued in 1908), you would only hold the note longer if there is a space equal or greater to one note's width between the note in question, and the following note on the same syllable (if there is one). Or if the note in question is right before a bar line.

- If you are following a nuanced semiological interpretation, you would consult the adiestamic nuems, as found in the Gradual Triplex, and make a judgement based on what sign is used for that note (perhaps also consulting Dom Cardine's book Gregorian Semiology).

- If you are following a measured/proportional approach, you would consult the oldest known manuscripts (probably mostly the same ones as are in the Graduale Triplex), and determine whether the note in question is long or short. The long notes are exactly twice the length of the shorts.

Other people may have their own set of rules, based on their own personal experience, what seems most "musical" to them, or just whatever is most intuitive, or the traditional practice in the group in which they sing.

Well, as the song says, one is the loneliest number, so it must be the tonic.

For those who sing Mass in the extraordinary form, there is a perfect example of the phenomenon on the second syllable of the word tenerent in today’s introit (Sunday in the octave of Christmas). According to the classic Solesmes method, this note is treated “expressively” with a bit of length.

That's a good question, which I think I can provide a helpful answer to! To be sure, I was a little imprecise/over-general in my comment above. Elsewhere you have made a useful distinction among several generations of the classic Solesmes method. Above I was using it in the sense of, say, the Denise Lebon/Fontgombault/ICKSP school rather than in the method as it was practiced in Dom Mocquereau's time.

But there is more to be said on the point. As you know well, Mocquereau says that he didn't choose to include everything from the manuscripts in his Solesmes editions, for a variety of reasons that have been aired at length elsewhere. But in his more theoretical writings, which we can distinguish from those methods, he does discuss this concept, so I'm pretty confident that he is the source of the practice and the particular terminology used within the Solesmes school.

The first place I know of his using the term désagrégation is in volume 11 of the Paléographie series, in his study on the notation of Chartres 47. And he uses it in precisely the sense we are talking about. On page 94, he writes:

If, for example, a copyist wanted to indicate that there should be a slightly prolonged emphasis on the first note [of a four-note neume low-medium-high-medium], and on that note only, how would he go about it? There's only one procedure available—disaggregation. That is precisely what we see in all three notations.

He gives as an example the classic case of the beginning of an eighth-mode tract, which is probably the most frequent place where the idea comes up in Mocquereauvian circles these days. Maybe he uses the term in an earlier book, too, but I can't think of an example right now. So I would say that the idea is present in the Solesmes method in Mocquereau's more semiology-leaning writings, even if it didn't make it into method books until later.

One is the loneliest number that you'll ever do
Two can be as bad as one
It's the loneliest number since the number one

I wonder if these guys wrote some of our more modern hymn

Charles,
If they did the hymns would sound much better than they do!

Francis, you made me look...
TDN = Three Dog Night for anyone else who didn't remember.