So let's say I write my own little script to create an ethereum transaction that transfers some amount to another address. I set a really low gas price. Then I sign it using a private key that has enough funds associated to make the transfer. Then I paste the transaction into the mainnet using etherscan.io.
If I understand ethereum well enough, if the gas price is too low, other miners won't (in the current proof of work situation) be motivated to do the work to mine that transaction.
But, if I have a synced node, could I cherry pick that transaction and do the work to mine it? If I were willing to forget about the benefit I gain from mining (the rewards in the form of gas), could I guarantee to some extent that the transaction would make it into the chain?
Is this likely to be expensive in terms of electricity? Is this something that would never make sense unless the transaction rake were significant such that I would make a large amount of money out of getting that transaction through.