CiviCRM 4.7.25 - No such field - unknown column a.weight in field list on Activity
Comparing a fresh CiviCRM DB with my own revealed the differences between existing and expected. 

I added the weight field to the civicrm_activity table in the right place and things were resolved. 

This installation has been through years and years of updates and there have been several occasions over the years where half-baked upgrade scripts have left the DB in similarly half-baked corners.

Gladly, this only caused some UI issues and is now resolved.