I dont think a lottery is a good idea but I believe that incentives to full node operators is a good idea. This is because full nodes are important members of the network.
I understand that there are currently enough full nodes on the networks. But the amount of full nodes cannot be guaranteed by goodwill alone. In order to guarantee that there is always a sufficient amount of full nodes an incentive is necessary.
The issue at hand is than, which full node shall incentives be given to.
The common issue with this scenario is that, it is easy for someone to game the system by running multiple copies of a full node. To combat this issue, we should give incentives to full nodes that are backed by x amount of bitcoin. The amount of incentivized full nodes will thus be limited by the total amount of bitcoin in circulation.