@Gregsaab, the good news is that this is a straightforward fix - The bad is that you're liable to find you need to remove the bridge from the guitar to do it, if there's corrosion or anything else necessitating elbow grease (I'll say a small prayer for your blood pressure if you try to fight it with the bridge still in the body). Basically, you need to tighten the threaded fastener nut on the other end of the vibrato arm bushing - The problem is that it's
round; meaning it will be tough to get a grip on the bushing from each side:
The two part bushing consists of a sleeve (with a button head on the collar-end, and threading on the other) inserted from the topside of the bridge, and a round fastener which threads onto the sleeve's protrusion on the underside of the bridge.
I decided to snap a photo of the disassembled parts for reference.
In your case, you need only thread the nut on more until the fitting is better secured; and the room for your symptomatic wobble is eliminated - For reasons I'll elaborate on below, be mindful to
not over-tighten it. The steps I came up with in my testing are:
1. Remove the set screw from the trem, and set aside in a safe place.
2. Insert a hex key (or the vibrato arm, if you can't find a key that fits) in the collar-end of the bushing, and
3. grip the nut end of the bushing with channel lock pliers or vice grips with a cushioning material between them to minimize scratches (in my case, an eyeglass cloth within reach was sufficient).
4. While squeezing the nut with the pliers firmly enough to prevent it from rotating, use the hex key inserted into the collar-end of the bushing as a crank to turn the bushing toward yourself (in the manner illustrated above) to tighten the assembly by threading the nut further onto the bushing sleeve (the opposite direction will, of course, unthread the nut).
5. Then, check the friction of the bushing fitting with the next step:
Using the vibrato arm inserted into the collar (with the set screw still removed), test whether the resistance the bushing is rotating against feels sufficient. If you tightened the nut too much, the bushing will not want to rotate; and you must repeat step 4, but turn the key in the opposite direction to loosen the fitting until you find you're happy with the amount of resistance.
6. Reinstall the arm tension set screw. Depending on how tightly you fastened the bushing assembly against the underside of the bridge, the resistance imparted by the set screw may be redundant unless you prefer the bushing loose enough for the bar to swing down toward the floor when not held (at which point the resistance imparted by the set screw can be used to compensate for the looser fitting of the bushing to an extent).