The Egyptians of the 26th-30th dynasties took a great interest in the tombs and artefacts of much earlier (1000-1500 years) dynasties, many of the inscriptions upon tombs and even the great pyramid which identify them were placed by priests and scribes who lived long after the builders of the structures. It is hard to name names of individuals, who did it, but there was at the very least one systematic attempt under Ptolemy II to catalogue Egyptian history from the buildings and temple records available.

I'd love to hear more about this if there are any Ptolemaic era specialists on the sub.

It's not exactly archaeology as in trying to gain knowledge of the past through excavation, but there is a story in Herodotus of the Spartans, during a war they were losing against Tegea, carried out a systematic search for the bones of Orestes. Relocating the bones to Sparta changed the course of the war, according to the story.

I think something closer to archaeology might be Herodotus himself, who, if you believe him, was investigating Egyptian monuments. I don't recall him doing any digging, but he claims to have got into underground chambers and such. But it's still closer to tourism than archaeology.

Some of the ancient Mesopotamian rulers dabbled in antiquarianism (the interest in and collection and preservation of old things because they are old, a precursor to the formal field of archaeology), most particularly King Nabopolassar (636-605 BC). According to Nabopolassar himself,

“When I was young, although the son of a nobody, I constantly sought out the temples of Nabu and Marduk, my patrons…shrines, walls and temples… which had weakened and collapsed because of age; whose walls had been taken away because of rain and deluge; whose foundations had heaped up and accumulated into a mound of ruins—I mustered Enlil’s, Shamash, and Marduk’s troops. I had them use the hoe and imposed the basket of conscription on them. From the bank of the Arhtu canal, on the lower side near the Urash gate, I removed its accumulated debris, surveyed and examined its old foundations, and laid its brickwork in the original place. I established its base on the edge of the underworld. I surrounded the east bank with a mighty mountainous belt….I Nabopolassar, the one who discovers (inscribed) bricks from the past, the one who implements the work on the original, eternal foundations, the one who wields the hoe of the Igigi.” (Nabopolassar Cylinder, emphasis mine).

As we can see in this inscription, which is not the only one Nabopolassar wrote about his activities restoring and recovering his civilization's ancient history, Nabopolassar engaged in several activities which are very similar to the things modern archaeologists also try to do.

He observed and noted the fallen condition of the buildings, employed (or conscripted) laborers to unbury the ruins (one of the earliest recorded excavations). Most importantly, he did not merely loot the ancient remains, but surveyed and recorded them, examining the foundations and performed restoration by laying brickwork in the original place. He also discovered ancient inscriptions and other artifacts, and made a note of these as well.

These efforts were undertaken to honor and redisplay the splendor of an earlier time, similar to what modern archaeologists do.

Some sources:

Diana Pickworth. “Excavations at Nineveh: The Halzi Gate.” Iraq [British Institute for the Study of Iraq] 67.1 (2005) 295-316, esp. fig. 1 on 296 and 298.

K. C. Hanson. Nabopolassar Cylinder, 2012. See Dr. K. C. Hanson, Wipf and Stock Publishers,199 W. 8th Ave. Eugene, OR 97401. See http://www.kchanson.com/ancdocs/meso/nabo.html

Here is a good magazine write-up about this:

http://www.electrummagazine.com/2012/01/king-nabopolassar-ancient-babylonian-archaeologist/