Quoted from BaxterStockman:Any recommendations for soldering irons? They all kinda look the same to me.
Seeing as how you only own 1 game - a new Stern TMNT, and you have a single wire on a coil that needs to be soldered on, I recommend getting a standard soldering iron, sponge, some solder, and maybe a solder sucker. A simple kit will let you make your repair then easily stash it away for the next time. There are plenty of operators and bar/arcade techs that make it happen with this level of tool. It simply plugs into the service outlet of the game. The Hakko is a wonderful piece of kit (It's the one that I have) for bench/board work and would be THE one to get if you had a larger collection with some older games that require more frequent soldering and board work but it is overkill for what you need right now. The power supply for it is awkward to place inside the game so I also have a simple soldering iron like this that I use when working on the games directly. That's my 2 cents.
https://www.amazon.com/Weller-SP25NUS-25-Watts-Soldering-3-000mm/dp/B00B3SG70K/ref=sr_1_4