You may have already heard about archaeologist Eilat Mazar’s work in Jerusalem. She has found some massive walls and structures that appear to date securely to the 10th century BC — Solomon’s era. You can read a summary of the story here, and take a look at her (Mazar’s) previous publication on the matter here.
For minimalists, there exists no evidence at all for the biblical Solomon. The Bible is ruled out as a propagandistic literary piece composed entirely after the exile. The finds at Megiddo, a heavily fortified city in the 10th century BC are also ruled out — redated by minimalists to the Omride period in Israel’s history. As a result, this find would speak to Solomonic building, but only because of the date and location (Jerusalem), not because anything with the name “Solomon” has been found therein.
You should also be aware that Eilat Mazar is a favorite target of minimalists — she is cast as an Israeli fundamentalist because she actually believes the biblical accounts. She gets accused all the time of sifting archaeology through the grid of the Bible (like the way archaeology used to in the Albright era). Nevertheless, she is the real deal — a real archaeologist of high pedigree as you can read here.