Electricity: Uhm, I'm an Eco major and high school has been a while ago. I barely remember that somehow magnetism is involved. I'm trying to limit whatever Stabby can introduce by not letting him be cleverer than I am.
You should play to your strengths: fund R&D, including the theoretical. It's a much more realistic route to future industrial revolution than most things from the genre of temporally displaced engineers anyway.
Knowing magnets are involved is already a big deal by itself, actually. Generating electrical current is as easy as moving a magnet through a spool of wire (heck, some of those rechargeable flashlights/torches that you have shake a magnet through a wire spool), and one of the key discoveries of the nineteenth century was simply noticing that an electrical current affects a compass needle. And you certainly have much interest in those given your naval ambitions.
This is not playing too smart because in reality, none of the above will be of any practical use to you by itself. While anyone playing with wire, magnets, and compass needles could easily figure out how to make a basic telegraph, the range of that sort of setup will not be anywhere near competitive with your semaphores, because long wires are both expensive and have huge voltage drop, so in practical terms you easily could make a telegraph to the next building, but not to the next kingdom.
But what you do have is a clear excuse of finding some bright people and giving them the task of figuring out how it works, with a huge advantage over history in at least knowing what's
possible, even if you have no clue whatsoever how to achieve them (e.g., batteries). You also (probably) know that science is driven by experiment and tests ideas by trying to experimentally prove them wrong, which is itself a huge innovation in thought.
Even with no industrial revolution within your lifetime, over the next generations Stannis Labs or Stormland Institute or whatever could easily become known as the birthplace of modern science. That's a neat legacy to have, and over the long term arguably easily eclipsing anything the Tywins of this world will achieve by itself (i.e., even separately from your other schemes). So if nothing else, do it for your ego.
Also, I think it may be interesting even on the short term—this is a potential of stepping on the Maesters' toes, but their monopoly on knowledge really does need to be broken.
