I don't yet have a reference, but it seems that the result was first proved by Fichtenholz and Sierpiński, independently from each other. This should be mentioned in a remark to Problem 12 in:
R. Sikorski, Real Functions, Vol. 1, PWN: Warsaw, 1958 (in Polish),
at least according to the historical remark on p. 28 in:
P. Lorenc and R. Wituła, Darboux property of the nonatomic $\sigma$-additive positive and finite dimensional vector measures, Matematyka Stosowana 3 (2013), 25-36.
Unfortunately, I couldn't retrieve either a hard copy of Sikorski's book or simply a scan of the relevant page(s).
Update (Nov 27, 2015). Thanks to Martin Sleziak and Jacek Jendrej, I found out that the "remark to Problem 12" referred to by Lorenc and Wituła is actually a footnote on p. 225 of Sikorski's book.
Now, Problem 12 reads, "If $\mu$ is a non-atomic measure and $0 < s < \mu(A) < \infty$, then there exists $B \subseteq A$ such that $\mu(B) = s$", so we are really talking of Theorem 1 in the OP. And the footnote on p. 225 really makes reference to "Sierpiński [7] and Fichtenholz 1". But the copy of Sikorski's book I could retrieve is incomplete and doesn't include the bibliography... Could anyone having access to a complete copy fill in this answer?
Update (Dec 07, 2015). Here is some more evidence that the common (?) attribution of Theorem 1 in the OP to W. Sierpiński may be wrong.
With the invaluable help of Nathalie Granottier and Fabienne Grosjean (Bibliothèque du CIRM, Institut de Mathématiques de Luminy) and Cyril Mauvillain (Bibliothèque de Recherche Mathématiques et Informatique, Université Bordeaux 1), I've finally got a scan copy of the bibliography of Sikorski's book.
The final outcome is that "Sierpiński [7]" (from the footnote on p. 225 of Sikorski's book) is nothing but the paper of Sierpiński that was already cited in the OP, namely
W. Sierpiński, Sur les fonctions d'ensemble additives et continues, Fund. Math. 3 (1922), No. 1, 240–246 (in French),
while "Fichtenholz [${}$1]" is
G. Fichtenholz, Sur les fonctions d'ensemble additives et continues, Fund. Math. 7 (1925), No. 1, 296–301 (in French).
Moreover, neither of these two papers deals with Theorem 1 in the OP, and this is not in contradiction to Sirkoski's statement (though it is in contradiction to all of the statements made in a bunch of papers attributing the result to Sierpiński and referring to the above paper of his), as the text in the footnote on p. 225 of Sikorski's book says "por. także", which sounds more like a "see also", which I interpret as a "cf.", than like a "see", which I would interpret as "the result appears in", and a "cf." makes perfect sense, indeed.
So, my conclusion is that, if anyone, then it's Sikorski who should deserve credit for the result, which in fact applies to countably additive measures (as of the statement in the OP), whereas measures considered in Sierpiński's and Fichtenholz' papers are finitely additive.
This has very much confused me in the first place, but the incipit of Section 3, Chapter VI of Sikorski's book reads:
§3. Miara. Zgodnie z uwagami wstępnymi z §1 miarą nazywamy każdą nieujemną, przeliczalnie addytywną funkcję zbioru $\mu$, określoną na pewnym przeliczalnie addytywnym ciele $\mathfrak M$ podzbiorów przestrzeni $X$, tzn. nieujemna funkcję rzeczywistą na $\mathfrak M$, nierówną tożsamościowy $\infty$, taką, że dla każdego ciągu nieskończonego zbiorów rozłącznych $A_n \in \mathfrak M$ $$\mu(A_1 + A_2 + \ldots) = \mu(A_1) + \mu(A_2) + \ldots$$
And this should more or less mean:
§3. Measures. Based on §1, we let a measure be any nonnegative, countably additive set function $\mu$, defined on a sigma-algebra $ \mathfrak M $ of subsets of $ X $, i.e. non-negative real function on $\mathfrak M $, possibly taking also the value $ \infty $, such that for every countable sequence of disjoint sets $ A_n \in \mathfrak M $ we have $$ \mu (A_1 + a_2 + \ldots) = \mu (A_1) + \mu (a_2) + \ldots $$
To wit, Sikorski is really talking of countably additive measures.