Encampment:+exp for land units/+production when building them
Harbor:+exp for land units/+production when building them/+gold for trade routes
Aqueduct: +food/housing
Neighborhood: +housing/+amenities
Airport: +exp for air units/+tourism
Spaceport: +science/+production for spaceship projects
I feel like they'd want to avoid tedious micromanagement. So something like +xp wouldn't work because I'd only want to work the slot on the turn I produce a unit, and no other time.
Going off of your ideas, I'll propose some.
Encampment - City defense/healing, city attack, and defensive bonus in the Encampment (you'd still only want to work this when the city is under attack, but that should be a broad 10 turns or so)
Harbor - +gold (or production or food) per trade route
Aqueduct - +(more than 1) housing. I can't imagine a Sewer system producing food
Neighborhood - If it has buildings, then +food. I'm imagining the buildings to be things like Mall, Supermarket, and Brewery
Airport - Tourism
Spaceport - I don't think these will provide specialist slots. The 3 buildings are only to build the 3 Science victory projects. You'd only build one per civ, right? Even if not, this is so late game that making it follow the other rules doesn't make much sense.
That also reminds me, I hope the Spaceport requires you to build it near the equator and not next to any other districts. You have to find a good spot for it!