I believe running a Lightning Network node, and charging for routing fees could be one model for incentivizing nodes to get some rewards from the network. 
Then it's incentive to run LN nodes, not Bitcoin full nodes since your LN client could act as SPV client and connect to another full nodes (whether it's owned by your or other user).
To install and run an LN node, you would also need to install bitcoind, unless you're using something like Neutrino. But if you are already maintaining a 24/7 full node, then you can install and run a Lightning node with it and charge for routing fees.