The National Archaeological Museum was recommended by our local tour guide, and turned out to be one of the highlights of Athens for me. The museum contains a wealth of ancient artifacts, statues, artwork, jewelry, and tools. The first main highlight was the Akroriri exhibit. After having seen the site itself in Santorini, we really enjoyed seeing some of the Murals and artifacts that were uncovered there. It really helped paint a complete picture of the place.
My personal favorite though was seeing the Antikythera Mechanism, the earliest known analog computer. It was used to calculate the dates of astronomical events, including eclipses, years in advance and dates to around 150 BC. As a software engineer, this was fascinating. I have seen documentaries devoted to the device, so seeing it in person was something special. In addition to the original pieces, they also have various reconstructions of the machine on display.