There was a pretty cool concept called "Proof of Activity" that - as far as I know - was never fully implemented. It was also the inspiration for some early Proof of Stake algorithms.
You can read the concept
in this whitepaper. You can see that there are some well-known names of the cryptocurrency community involved (like Charlie Lee).
Basically, the algorithm selects randomly a satoshi from all satoshis that are spendable (contained in UTXOs) at a given time, and looks if the owner of the satoshi is currently online (he must solve a small task to prove that) - if not, then he is not rewarded. That's also why the algorithm was called "Follow the Satoshi". People that own more Bitcoins, in this concept, have more chances to get selected, so it's actually a "proof of activity/stake hybrid".
Another coin that rewarded full nodes was Timekoin, but it had a completely different algorithm than Bitcoin.
This is an amazing whitepaper and IT DOES solve several of the problems that BTC faces today. Any idea as to why this project did not manifest? Any news regarding a potential revival?