I've heard that running a monero, or any alt-coin node can gain rewards for something. Blockchain discovery maybe? What is this and how does it get claimed?
Normally, just running a full node (i.e. a node that contains a full copy of the blockchain) just entails keeping your copy of the blockchain up-to-date, and also relaying new transactions. These transactions are then included in the blockchain by the process of adding a new block to it, know as mining. However, to get the privilege of editing the blockchain by augmenting it by this new block (which will include the new transactions), the user's software has to perform intense computations in the hopes of beating the other nodes to solving a puzzle. If he does, then he gets the block reward, which is currently around 10 XMR per block.
So you can think of mining as a race to win a lottery that has a prize of about 70 USD every 2 minutes, and the odds of winning this lottery will depend on the hardware and energy the user decides to invest into it.
Mining is optional, and there isn't a reward for just relaying transactions or keeping an up-to-date copy of the blockchain; not in Monero at least.