I can make some recommendations for soldering tools - I’ve taught a couple hundred people soldering through workshops with a local maker space, so have some opinions on the equipment that’s available 
Soldering Irons and Stations
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Go with a temperature controlled iron.
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Temperature controlled irons let you get a more powerful iron, and not worry about it burning itself out, or overheating and causing issues with nearby components. Power supplies have gotten a lot smaller, so it’s very possible to get a portable sized iron (no base station) that’s got plenty of power and temperature control.
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More powerful irons heat up hot enough to melt solder quicker, so with no risk of overheating, they’re much more convenient.
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Temperature controlled stations have gotten cheap enough that it’s just not worth going with un-regulated ones.
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The Hakko FX-601 is an awesome portable iron - temperature controlled, 67 Watts. It doesn’t come with a stand or a sponge. I think for a portable pinball toolkit it’s the best thing out there.
For more traditional solder stations, the pretty common Weller WES-51 is only 50W - less powerful than the FX-601 (67W), so you may not feel the need for a more traditional station any more. The advantages are nicer ergonomics of the pen and a lighter cord, but the base is inconvenient for pinball work.
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If you are going to buy one, I like the Weller WES51/WES51D or the Hakko 888D in the (~100) price range, individual preference between the two.
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If you want cheap, there’s a number of companies selling clones of the old Hakko 936 under a variety of different names (Auyue and Yihua being two notable ones).
Hobbyking, who are the biggest online hobby RC supplier around have it for the cheapest -~$16. It’s a reasonably competent 60W base station, and the price is hard to beat. They aren’t the most reliable, or the nicest to use, but they sure are cheap. About the same size as the Weller WES51.
- If you want to spend more than $200, Hakko, Weller, Metcal, Pace, JBC are all reputable brands that make excellent systems, but the benefits are more for people doing small precision work, not so much for pinball repair.
- Replacement tips are quite affordable for most common stations.
- Replace the tip if solder stops adhering to the tip nicely when you solder.
- Don’t scrape away at the tip with something sharp - it’s an old-school technique for old-fashioned iron design, and doesn’t do well with more modern tips where the outer skin of the tip is a different material plated on. At the point the plating is pitted enough to not be performing well, scraping isn’t going to help it.
Portable De-Soldering Tools
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Many people like solder sucker tools - small spring powered reverse syringes where you heat up solder with an iron, then stick the tip of the solder sucker in it, and press a button to suck it up.
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I’m not a huge fan of solder suckers, as I find the results inconsistent, and they often leave bits of solder that are difficult to remove, as if the tip of the sucker can’t fully sink into solder, it won’t pick much up.
I prefer desoldering braid (AKA wick) - like this stuff. you put the braid against the solder you wish to remove, then hold an iron up to the braid. As the solder melts, it’s wicked into the braid. Much cleaner.
- Adding flux to something tricky (big wire, board work, etc.) helps get good results - for desoldering braid, putting some extra on braid (especially old braid) helps the wicking.
Removing components
- If you’ll be doing a lot of removing through hole components like driver MOSFETS, a desoldering station can be convenient -
- these are like a combination solder sucker and soldering iron - the iron has a hollow tip, and a motorized vacuum is used to suck the solder out as it liquifies. Hakko makes a popular one, the FR-300, and there are other more expensive manufacturers out there too.
- There’s a guy on pinside who apparently offers pinball people a discount on these, but I haven’t followed up on that.
I’d however suggest a different option - a hot air pencil.
- These are basically the hair dryer from hell, and the cheap Chinese brands are good value, and perfectly functional -
- they work by blowing a fine amount of air, heated enough to melt solder.
- You use it by dancing it across the leads of the part you’re trying to remove until the solder melts, then pull the part out of the board with tweezers.
- An advantage is that multiple leads can be heated simultaneously, which makes it easier to remove parts with more than one or two leads (chips, drivers, etc.). Once the part is removed, use desoldering braid to clean up the solder.
- Disadvantage is risk of heating up parts you don’t intend.
Hot air pencils are also really handy for surface mount components (anyone had to do any board work on Stern’s Spike system stuff yet?).
Solder
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Don’t use lead free solder -
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It’s much more annoying to work with, and as long as you wash your hands after working, leaded stuff is fine.
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Don’t use solder or flux intended for plumbing - it’s far more acidic, and can damage boards and chips.
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You want stuff labeled as “Rosin Core” -
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it’s a small amount of flux embedded in the solder wire, that burns off while soldering to prevent oxidation, giving you a better solder joint.
“Pocket Packs” are a narrow test-tube sized plastic container of solder with a hole on the top to let solder out. It’s a convenient way to carry it in a portable toolkit, but more costly. Many vendors sell in this form factor.
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Two major reputable solder suppliers in North America are MG Chemicals (Canadian, sells in US and Canada) and Kessler (American).
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Some people swear by eutectic solder (AKA 63/37), a marginally more expensive leaded solder alloy than the standard 60/40. It’s marginally nicer to work with, but not the big leap of any leaded over non leaded solder.