Digging through the old photos and records for changes
And another form of astronomical archaeology is checking old photographic plates and records of celestial events to see what may have changed with a particular object over time.
Of course some day we may also be doing actual space archaeology too:
http://www.archaeology.org/0411/etc/space.html
What vessels and probes would you like to see found and examined? My first choices would be some of the old Soviet lunar and Mars probes to see what happened to them. The Veneras would also be interesting, but probably a bit more difficult to reach for now.