Well here is one answer: The study of the Gauss-Kuzmin problem eventually led to a fruitful connection between continued fractions and functional analysis. Namely, the distribution function of the Gauss measure is the leading eigenfunction for the transfer operator associated to the Gauss transformation. It turns out that this is the hidden explanation for the theorem you quoted in your question.
Understanding this connection recently led to a detailed study of transfer operators of the Gauss map and related transformations, and their associated Dirichlet series, which paved the way for major breakthroughs by Baladi, Vallee, and others, in understanding the statistics of the Euclidean algorithm and its various analogues. It would be difficult to give more details than this without writing a very long post, but if you are interested in finding out more then here are two references:
1) "Euclidean algorithms are Gaussian", Baladi and Vallee, available on Viviane Baladi's webpage.
2) "Continued fractions", Doug Hensley, Chapter 9.
I think part of the answer may be found by consulting Volume X of Gauss's Werke. "REV. GALEN" doesn't actually appear in the Tagebuch itself, a facsimile of which appears following page 482. It was jotted down by Gauss elsewhere, as explained on page 539, in the commentary (which runs for nearly three pages) on the Tagebuch entry dated April 8, 1799.
Just above the excerpted paragraphs from Men of Mathematics, Bell writes, "A facsimile reproduction [of Gauss's diary] was published in 1917 in the tenth volume (part 1) of Gauss' [sic] collected works, together with an exhaustive analysis of its contents by several expert editors." I think it's safe to assume that Bell actually looked at this 1917 publication (and I think it's reasonable to assume that the 1973 edition I'm looking at right now is not substantially different), and I think it's fair to conjecture that Bell paid more attention -- but maybe not enough! -- to the transcription and commentary than he did to the facsimile.
As for the misdating of "Vicimus GEGAN," the correct date is clear enough in both the facsimile and in the transcription on page 507. For one thing, it appears immediately below an entry dated October 18. My guess is that either Bell or the typesetter made a simple mistake.
Finally, a useful reference, especially for "GEGAN" (and a related notation, "WAEGEGAN") is Mathematisches Tagebuch : 1796-1814. Unfortunately, my command of German is insufficient to give a good synopsis of what's to be found there. I hope an actual historian will weigh in here.
Added Feb. 21: It turns out there is a 2005 edition of Mathematisches Tagebuch 1796-1814 (the copy I found earlier was a 1985 edition) which has an update referring to a 1997 paper by Kurt Biermann. Here is a relevant Zentralblatt review of that paper:
Zbl 0888.01025
Biermann, Kurt-R.
Vicimus NAGEG. Confirmation of a hypothesis. (Vicimus NAGEG. Bestätigung einer Hypothese.) (German)
[J] Mitt., Gauss-Ges. Gött. 34, 31-34 (1997).
The author, a well-known expert on Carl Friedrich Gauss, reports on a Gauss-manuscript, which was found recently in the Göttingen astronomical observatory by H. Grosser and which confirms a hypothesis by Biermann from 1963. At that time Biermann read the frequent code GEGAN in Gauss' diary and manuscripts in inverse order as standing for (vicimus) N[exum medii] A[rithmetico-] G[eometricum] E[xpectationibus] G[eneralibus]. This in turn was alluding (in Biermann's opinion) to Gauss' discovery of the connections between the arithmetic geometric mean and the general theory of elliptic functions. The recently found Gauss-manuscript shows, for the first time, the code NAGEG, and, on the same sheet (which is reproduced in the article), the well-known GEGAN alongside with the picture (by Gauss' hand) of a lemniscate. Thus a remarkable historical hypothesis has been essentially solved after more than three decades.
[R.Siegmund-Schultze (Berlin)]
Best Answer
The entry from May 24, 1796 is worked out in a more general form on February 16, 1797 [reproduced below from this scan]
$$1-a+a^3-a^6+a^{10}+\cdots=\frac{1}{\displaystyle 1+\frac{\strut a}{\displaystyle 1+\frac{\strut a^2-a}{\displaystyle 1+\frac{\strut a^3}{\displaystyle 1+\frac{\strut a^4-a^2}{\displaystyle 1+\frac{\strut a^5}{1+\cdots}}}}}}$$
so the coefficients alternate between $a^{2n+1}$ and $a^{2n}-a^n$.
Latin text: Amplificatio prop[ositionis] penult[imae] p[aginae] 1, scilicet $\cdots$
Unde facile omnes series ubi exp[onentes] ser[iem] sec[undi] ordinis constituunt transformantur.
Translation: Expanding on the proposition 1 from the next-to-last page $\cdots$
From here one can easily transform every series the exponents of which form a series of the second order.
These continued fractions of series of the form $$1+\sum_{n=1}^\infty a^{n(n+1)}-\sum_{n=1}^\infty a^{n^2}=\sum_{n=0}^\infty (-1)^n a^{n(n+1)/2}$$ are related to theta functions, see chapter 29 of "Series and Products from the Fifteenth to the Twenty-first Century". Apparently the series originated from Jakob Bernoulli (1690), see History of Continued Fractions and Padé Approximants.