Where would the funds for these rewards come from?
Who would be responsible for distributing them?
How would you determine how much reward any given node should receive?
Have you considered that this might "centralize" nodes... Encouraging large clusters of full nodes to claim rewards?
The reward of running a node is helping to keep the network decentralized and secure.
To expand on HCP's post: Assuming one would reserve a portion of the block subsidy to reward people running full nodes you'd still have the problem of distribution. Problem being, while the work aspect of PoW can not be "faked" (ie. the amount of computational power required is cryptographically secured),
pretending to be a multitude of full nodes is fairly trivial (eg. on the extreme end an ISP could simply point hundreds or thousands of IP addresses to a single machine and nothing would be gained in terms of decentralization or reliability). That is to say, a full node subsidy is much easier to be gamed than a PoW-based mining subsidy.
It is also worth noting though that the original design of Bitcoin doesn't seem to have accounted for the division of mining and non-mining full nodes we see today (ie. the first versions of the Bitcoin reference client included the mining logic as well). However there's a lot of speculation in terms of what satoshi did and did not intend so I'll leave it at that.
