What was the first solar eclipse that was demonstrably predicted in advance?

I am going to second dotancohen's answer somewhat. Hipparchus developed a comprehensive astronomy that accurately predicted eclipses and other astronomical events. Ptolemy's writings emanate from the tradition that was established by Hipparchus.

Nevertheless, Hipparchus was certainly not the beginning of Greek astronomy. He simply formalized and improved it. Long before Hipparchus many Greek philosophers were very capable in astronomy and mathematics, such as Thales of Miletus. Although Herodotus lived 100 years after Thales, there is no specific reason I know of to doubt his claim that Thales predicted his eclipse.

Herodotus says that Thales "discovered" the mechanics of eclipses, so this would suggest he considered this new technology at the time, improvements over the Egyptian and Babylonian methods. It is known that the Babylonians could predict lunar eclipses which is pretty easy, but not solar eclipses and the same is probably true for Egyptians.

Also, remember that ancient scientists had access to lots of writings that are now lost, including material on Thales. None of these contradicted Herodotus or corrected his account. In fact, later Greek astronomers repeatedly confirmed what Herodotus had claimed which is that Thales was the first to do it.

On July 30, 2014, Tony Freeth published his findings in PlosOne, that ancient Greeks were predicting lunar eclipses using the Antikythera Mechanism some time between 250BC and 1BC. Aside from predicting dates, this remarkable device was also able to predict magnitude, color, and obscuration.

This detailed article argues for the authenticity of Herodotus' report about Thales ecplise prediction in 585 BC. This is in any case a lot earlier than the Chinese material cited by Semaphore.

From a NASA answer:

Ptolemy ( ca 150 BC)[sic] represents the epitome of Greecian astronomy, and surviving records show that he had a sophisticated scheme for predicting both lunar and solar eclipses. Ptolemy knew, for example, the details of the orbit of the Moon including its nodal points, and that the Sun must be within 20d 41' of the Node point, and that up to two solar eclipse could occur within seven months in the same part of the world. Lunar eclipse were especially easy to calculate because of the vast area covered by the Earth's shadow on the Moon. Solar eclipses, however, required much greater finesse and knowledge. The shadow of the Moon on the Earth is less than 100 kilometers wide, and its track across the daytime hemisphere is the result of many complex factors that cannot be anticipated without a nearly-complete understanding of the lunar orbit and speed.

The writeup (available here) also supports the other answers here and provides a lot of related information. (note that Ptolemy actually lived AD, not BC)

Wikipedia has an informative article on the Saros cycle, which is used to predict eclipses.

According to that page, and by extension apparently the pages to which it references, the Babylonians were recording the eclipses which describe the cycle in the sixth century BC. Apparently Hipparchus (second century BC), Pliny (first century AD) and Ptolemy (second century AD) were aware of the cyclic nature of eclipses, though the degree to which they could be predicted for a specific location is debatable.

(This is an incomplete answer since I don't know which eclipse specifically was predicted, nor how it compares to the rest of the world. But it is too long for a comment.)

Because of their cultural association of governmental legitimacy with astronomical/geophysical omens, ancient China was rather obsessed with predicting eclipses. Attempts to do so seemed to have begun in the Warring States era, but naturally were not very successful. Some breakthroughs were achieved in the Eastern Han when it was realised that the moon's motion is inconsistent.

By 20 B.C. the Chinese knew how eclipses were caused ... By 8 B.C. the Chinese could predict eclipses by using the 135 month period; and by A.D. 206 they could predict eclipses by analyzing the motion of the moon. By A.D. 390 they could predict how much of the moon would be in shadow.

_{- Thurston, Hugh, Early Astronomy, Springer, New York, 1994}

These are referring to solar eclipses, as was made clear by the rest of the paragraph discussing ancient disputes over how the moon could block the sun's light. See also:

Astronomers were attached to the royal household as second tier functionaries ... One of the most important events to predict were eclipses. In the first century B.C. (the Han dynasty), an eclipse period of 135 months was recognized during which 23 eclipses were known. By the third century A.D., the astronomer Yang Wei was able to specify times of first contact for a solar eclipse.

_{Case Western Reserve University: Journey Through the Galaxy-}

By about 20 BC, surviving documents show that Chinese astrologers understood what caused eclipses, and by 8 BC some predictions of total solar eclipse were made using the 135-month recurrence period. By AD 206 Chinese astrologers could predict solar eclipses by analyzing the Moon's motion.

_{- National Aeronautics and Space Administration: Eclipse Through Traditions and Cultures}

It would therefore appear that, that by about the early third century, at least some eclipses were predicted.

Of course, these predictions were not very good, especially since the sun's movement had not been understood. That happened much later in the Tang Dynasty. The monk Yi Xing was able to produce better eclipse predictions based on his research. Another notable advancement in Chinese astronomy was achieved in the Yuan Dynasty, by the astronomers Wang Xun and Guo Shoujin.

This actual movement of the sun was fully understood by the famous astronomer Yi Xing (一行, 683-727 A.D), and he used the theory in his computations, for example the calculations for the time fo eclipses, in his Da Yan calendar (大衍歷, 729 A.D), getting accurate results in the process.

(...)

Guo and his co-workers were able to make more accurate astronomical calculations, which helped them to make better astronomical predictions, especially in the case of eclipses.

_{- Tiong, Ngsay, and and Helmer Aslaksen. "Calendars, Interpolation, Gnomons and Armillary Spheres in the Work of Guo Shoujin (1231–1314)."}

Again, the eclipse predictions were still not perfect, especially over the course of centuries. However, a failed prediction was cause for commissioning a new calendar. Conversely, this meant that at least some of the predictions even in this early period were accurate, by chance or otherwise.

One major piece of evidence proposed by Xu in favour of the adoption of Western methods concerned eclipse predictions ... [E]clipse prediction was the touchstone. In 1610, when it was first proposed to employ Jesuits for astronomical reform, the miscalculation of a solar eclipse was used to make the case for the necessity of this reform.

_{- Jami, Catherine, et al, eds. Statecraft and Intellectual Renewal in Late Ming China: the Cross-Cultural Synthesis of Xu Guangqi (1562-1633). Vol. 50. Brill, 2001.}

At this juncture the then-extant Chinese calendars were failing in their predicative powers. The Court held a competition (to predict an eclipse) between the Chinese court astronomers and the Westerners, which resulted in a resounding victory for European astronomy.