My client would like to report or export certain activities for contacts that meet a certain criteria. This doesn't seem possible, which is surprising. Activity search only allows a search of activity data - not contact data so contacts can't be filtered there although contact data can be exported from there. Advanced search allows you search on contact and activity data, but then only allows an export of contact data - so no activity data. It doesn't seem possible to have both contact AND activity data exported after filtering on both. Or have I missed something?
When I tried in the demo site, I did an Advanced Search->Display Results as Activities and filter by the field you want and search.
If you export the result and check either Export primary fields or Select fields for export, in the csv created will be the contact and activity information. You will see Activities in the lasts columns.
If you tried that and doesn't work it's posible that in your version you can't do that, but I doubt it.