Big Daddy Missions :
Big Daddy Repair Mission - find spare parts (or machines to make parts) in a cache identified by records
Another - would be (for a change) to help a Big Daddy instead of kill it. Go to it to make repairs/rescue it.
Now you still might have some trouble communicating the fact you are there to help it (and some difficult missions may force you to overpower it, if it is particularly deranged/out of communication).
Big Daddies were still being created fairly late by various factions (the availability of mind-gone Splicers was never a problem). At least some people realized that maintenance had to still get done. The group at Hephaestus probably made a deal with remnants of Fontaines people in Point Promethius to keep their Power & Heat on in return for properly programmed Maintenance Cyborgs (their original name) to work throughout the City. The Union of Fishermen in Neptunes Bounty probably were part of the deal to supply food as well.
At Central Computing (Maintenance Central - a Thinker sub-system) assigns maintenance tasks to the Maintenance Cyborgs, and they are usually the ones that detected their own problems and made a request to 'Central' (as programmed) to get their required repairs. Many of the old maintenance Depots are running short of parts and personnel.
Prestige Points are usually earned for assisting the City's repair infrastructure in this way. Similar Missions can include delivering materials or special equipment to some Big Daddy in a remote location to assist a particular repair (frequently it is an 'urgent' repair situation).