These are the current things I have thought of that can be invested in for each system. Note, these don't include any military assets except bases. Stuff like Ground-to-Orbit weaponry will be covered in Military.
Assets cost a certain amount to build and many have an upkeep that must be paid each turn. Many Assets require some Technology to be reseached before they can be built.
Industrial Complex: Increase Production by 1 per level. No Upkeep.
Cloning Facility: Requires Cloning Tech. Cost to increase Population is decreased by 25%
Research Faciltiy: Provides 1 Tech per level
Religious Santuary: Each level increases Max Morale by 1
Military Base
Orbital Base
Shipyards
Orbital Shipyareds: Increased dockspace for construction
Suppy Depot: Each level increases Supply Rating by 1
University
Trade Center
Terrformers: Required Terraforming Tech. Each level increases capacity by 1.
Robotic Facility
System Capital - Decrease Base Morale Modifier from distance to homeworld by 10%. Directly adjacent system enjoy a decrease of modifier by 5%. Limit 1 per Empire.
Automated Factory: Rquires A.I. and Total Autonomation Techs. Add the level of this facility to Pop when determining Production Capacity. Each level also inflicts -5% Base Morale
Hyper-Relay Station: Requires Interstellar Communication Tech. Halves Base Morale Penalty for being distant from homeworld.
Suggest more, please.