Hilbert's 23 problems, ten of which were presented at the 1900 ICM in Paris, are too famous for any mathematician to not know. If one reads the descriptions of the problems in Hilbert's paper, one realizes that some questions are concrete whereas the others are stated somewhat vaguely. The 24th problem that I will quote below definitely falls into the latter category.
It seems that there was a 24th problem which was "cancelled". The following is from an article that appeared in American Mathematical Monthly in 2003.
Let me begin by presenting the problem itself. The twenty-fourth
problem belongs to the realm of foundations of mathematics. In a
nutshell, it asks for the simplest proof of any theorem. In his
mathematical notebooks [38:3, pp. 25-26], Hilbert formulated it as
follows (author's translation):The 24th problem in my Paris lecture
was to be: Criteria of simplicity, or proof of the greatest simplicity
of certain proofs. Develop a theory of the method of proof in
mathematics in general. Under a given set of conditions there can be
but one simplest proof. Quite generally, if there are two proofs for a
theorem, you must keep going until you have derived each from the
other, or until it becomes quite evident what variant conditions (and
aids) have been used in the two proofs. Given two routes, it is not
right to take either of these two or to look for a third; it is
necessary to investigate the area lying between the two routes.
Attempts at judging the simplicity of a proof are in my examination of
syzygies and syzygies [Hilbert misspelled the word syzygies] between
syzygies [see Hilbert [42, lectures XXXII-XXXIX]]. The use or the
knowledge of a syzygy simplifies in an essential way a proof that a
certain identity is true. Because any process of addition [is] an
application of the commutative law of addition etc. [and because] this
always corresponds to geometric theorems or logical conclusions, one
can count these [processes], and, for instance, in proving certain
theorems of elementary geometry (the Pythagoras theorem, [theorems] on
remarkable points of triangles), one can very well decide which of the
proofs is the simplest. [Author's note: Part of the last sentence is
not only barely legible in Hilbert's notebook but also grammatically
incorrect. Corrections and insertions that Hilbert made in this entry
show that he wrote down the problem in haste.]
The paper I linked above discusses the history and the role of Hilbert's problems and I think is worth reading. Most of mathematical logic, as we know it right now, did not exist when this question was asked and you can simply disregard the question by saying "this is not a mathematical question". On the other hand, the same could be said about the second problem on the consistency of arithmetic today, if mathematicians did not develop the necessary tools to deal with this problem.
My point is that one might be able to answer Hilbert's 24th problem if one finds the "correct" statement of the problem. With our current knowledge and understanding of mathematical logic, can we define a criteria for a proof to be "simple"? Have there been any attempts to define such a notion? Should "simple" merely mean "short"?
In Thiele's article, you can find some quotations in Section 5 but they do not really give any useful information about how Hilbert perceived the word "simple". Having stumbled upon this article only today, I admit that I have not searched for other articles yet. So I would also appreciate being directed to other books and articles on this cancelled 24th problem.
Best Answer
some recent contributions to Hilbert's 24th problem:
Towards Hilbert’s 24th Problem: Combinatorial Proof Invariants, D.J.D. Hughes (2006)
Hilbert’s 24th Problem & Computer-Assisted Formalized Mathematics, B.W. Paleo (2014)
Hilbert’s 24th Problem, Proof Simplification, and Automated Reasoning, L. Wos
the earliest work on Hilbert's 24th problem is by Gerhard Gentzen (1933); it was discussed in Logic's Lost Genius: The Life of Gerhard Gentzen, by E. Menzler-Trott:
Gentzen's doctoral thesis "Investigations into logical reasoning" from 1933 was lost, and only rediscovered recently. (The main results were reinvented by D. Prawitz in the 1960's.) An English translation from 2008 can be found here: Gentzen's Proof of Normalization for Natural Deduction.