Create and maintain a list of all the bitcoin nodes that have been active for some period of time, maybe a year or some other time period. Node owners must have a certain amount of time invested before becoming eligible for payments.
Impossible to do in a decentralized way.
For starters all the full nodes can not know about all other full nodes ever, they also can't continuously check whether all the candidates in that list are active at all times.
Most importantly how are you going to distinguish a fake full node with a real one? It is actually very easy to fake being a full node without actually running one. If you provide the incentive, the network could be flooded with them.
As each block is completed, select one node, maybe at random, and the owner of the node gets 0.38 BTC. Take that node off the list to received block rewards for some period of time to allow for sharing the wealth.
This sounds like changing (increasing?) the total supply which is not something that the community would accept.