We should note that the underlying evidence behind the spiced up modern titles like "seat of the mind, "seat of intelligence", etc., is generally very obscure, and Wikipedia's phrasing is copied almost verbatim from an old and sketchy paper of Gross. SEP's recent account is more ambiguous as to what it was that Alcmaeon thought the brain did:
"There is no explicit evidence, however, as to what Alcmaeon meant by understanding. The word translated as understanding here is suniêmi, which in its earliest uses means “to bring together,” so that it is possible that Alcmaeon simply meant that humans are able to bring the information provided by the senses together in a way that animals cannot (Solmsen 1961, 151)... It is possible that we should use a passage in Plato’s Phaedo (96a-b = A11) to explicate further Alcmaeon’s epistemology. The passage is part of Socrates’ report of his early infatuation with natural science and with questions such as whether it is the blood, or air, or fire with which we think. He also reports the view that it is the brain that furnishes the sensations of hearing, sight, and smell. This corresponds very well with Alcmaeon’s view of the brain as the central sensory organ".
But even the brain as the sense central might have been a superficial speculation. In any case, it failed to convince most Greeks who continued to designate the heart as "the seat", following the Egyptians before them:
"Alcmaeon’s conclusion that all of the senses are connected to the brain may have been drawn from nothing more that the excision of the eye and the general observation that the sense organs for sight, hearing, smell, and taste are located on the head and appear connected to passages which lead inward towards the brain... His identification of the brain as the seat of human intelligence influenced Philolaus (DK, B13), the Hippocratic Treatise, On the Sacred Disease, and Plato (Timaeus 44d), although a number of thinkers including Empedocles and Aristotle continued to regard the heart as the seat of perception and intelligence."
At this level of detail and confidence, there are earlier mentions that can be interpreted as assigning "thinking" to the brain in the ancient Mesopotamia (c. 2000 BC). For example, after a long interpretive analysis of the sources, Goodnick and Sigrist conclude in The Brain, the Marrow and the Seat of Cognition in Mespotamian Tradition:
"In conclusion, according to our argument the Sumerians noted the existence of the brain, which they designated ugu-dig, and believed that understanding passed through the ear to the seat of the intelligence. On the other hand, the Akkadians not only collapsed the lexical distinction between the Sumerian lemmas, ugu «cranium» and ugu-dig "brain" but also equated both with the Semitic muhhu «marrow» (of bones including the cranium). The ancient physicians did not know the function of the brain as an organ and it is quite likely that they considered it a type of skull marrow. Nevertheless, the ancient metaphysical interpretations, both Sumerian and Akkadian, place understanding, reason and wisdom in the ear. Thus, despite the heart being seat of will, the head or brain, the place where temu «reason» is found, must be the seat of cognition, whether or not the medical profession understood the functions of the brain as an organ."
A comprehensive review of the early history, with analysis of the sources, including the Smith and Ebers papyri, is the collection History of Neurology. Minds Behind the Brain: A history of the pioneers and their discoveries. According to York and Steinberg's contribution, even with Egypt we have to interpolate:"cavalier treatment of the brain suggests that the ancient Egyptians had no real conception of brain function, or of the primacy of the brain in cognition, though certainty on the subject is not possible". The earliest place that offers something more cogent and definitive than speculative interpolations is the aforementioned Hippocratic treatise On the Sacred Disease (c. 400 BC), discussed by Karenberg:
"I hold that the brain is the most powerful organ of the human body, for when it is healthy it is an interpreter to us of the phenomena caused by air, as it is the air that gives it intelligence. Eyes, ears, tongue, hands and feet act in accordance with the discernment of the brain... Wherefore I assert that the brain is the interpreter of consciousness... Men ought to know that from the brain, and from the brain only, arise our pleasures, joys, laughter and jests, as well as our sorrows, pains, griefs and tears. Through it, in particular, we think, see, hear, and distinguish the ugly from the beautiful, the bad from the good, the pleasant from the unpleasant..."
A more popular recent history text is Finger's Minds Behind the Brain.