Make sure your return spring isn't too tight. Also, I have found filing the coil slug can lessen buzzing quite a bit. Finally, make sure the activator plate is installed correctly and is flat to the coil when pulled in.

The armature plate must be clean and square to the coil lest noise occurs.

You can swap the coil armature with another that doesn't get used often, such as the coin relays. The noise is an indication of wear on the armature, which you'll see once removed.

I had a really bad buzzer on my Williams Ding Dong. I think it was the ball index relay-as soon as you started another ball, that thing was humming like crazy. I filed it down and it's been quiet ever since.

Can't hurt to give it a try.

That Coil looks pretty toasty. Did you check it with an OHM meter and compare it to a like coil to make sure it is not going bad? Could be as simple as getting a new coil.

Ken

As I have gotten my first em, and it has a couple noisy/buzzing relays I also have some dumb questions.

How do you actually change the spring tension? I assume you don't want to just pull on it, thus weaking the spring overall? Do you need to buy a different tensioned spring?

Judging from the solder, I'd say that coil's been replaced or removed once before. Check the schematic and make sure it's the correct coil for the spot. But a new coil and armature plate, with correct spring, should do the trick.

If you've got a coil that's on for very long, you need a higher resistance coil so it doesn't burn up. An alternating coil shouldn't be on that long, but that one, as noted, looks pretty toasted. It's not so much a matter of work/doesn't work, but a matter of the heat generated by the coil staying on. Lower resistance is going to get hot and burn up faster than a higher resistance.

Since the buzzing comes from non-flat surfaces, usually the armature, and you already swapped the armature for a better conditioned one in the game, you're left with little alternative, (no pun intended).

replace coil, noise goes away...had a 1st ball relay in a jet spin that was toasty and buzzing, replaced/fixed

I think it may be a mistake to overgeneralize. You need to think about why the relay is buzzing. As the AC moves through zero, the relay opens up a bit, then it re-cinches up as the current moves off zero. This happens because the relay isn't strong enough to hold the armature on. Why? Either the relay is too weak, or the pull off force is too high. Reasons for the weak force? Poor condition coil, low voltage for one reason or another, poor condition of the "slug", armature too far from the coil. Reason for too much force? Too heavy a spring, too much tension on the blades, other mechanical reasons.

After looking at the poor condition of the coil, this is where (correctly) most posters went on this problem, but it may not be applicable to other relays. At $10/try, you may want to be sure before firing new parts in.

If anyone could follow up to how do you change the resistance of the spring I would appreciate it. Looking at my coils that are buzzing I don't see anywhere else other than where it is now, to hook the spring to. So to loosen the tension to you physically pull apart the spring and thus weakening it, or do you have to buy a less tense spring?

I usually swap out the spring for another, weaker one. I typically uses the "latch" spring for a gottlieb drop target.

It may be that the moving switch blades are adding tension to that which the spring applies to the armature. I don't think Gottlieb specified the wrong springs, of course, they may have been substituted sometime in the past. It is always worth spending a bit of time making sure the moving switch blades exert minmum pressure on the ladder so they do not add excess pressure.
The image shapeshifter posted showed the armature had gone past being a full mating to the coil slug end, and wasn't sitting on it fully. I suspect the new, larger coil cured that problem by making the fit better. A slight modification of that armature could well have cured the problem as well.