Solenoid

As Cats said, the black box is the holder for the bike's main fuse but I believe the main fuse on CX500s up to 1981 is 20A, not 30.
It would work if you bypassed the main fuse but it is not a very good idea because it is there to protect the wiring, alternator and regulator/rectifier from excessive current if something goes wrong.

The newer type that Mike linked to could be used but you should compare the wiring drawings for the CX500 and the GL500 to figure out how to connect it.

Welcome to the forum. Please add your location and your bike's model and model year to your profile so that you don't have to remember to tell us every time and we don't have to keep asking when you forget (see Forum Settings link in my signature).

And welcome to the world of antique vehicle ownership (they own us, not the other way around). Your bike is about 4 decades old and may or may not have had all of the maintenance necessary to keep it safe & reliable so it is highly recommended to download the Factory Shop Manual for your model (available through the CX Wiki - link in my signature) and go through all of the service procedures, regardless of whether your bike has reached the specified mileage.
Note that while aftermarket shop manuals are pretty much necessary for people without factory training to work on a lot of makes & models of bike the FSMs for the CX/GL500/650 family of bikes are so well written & laid out that the FSM is really the only book you need and and the aftermarket books are secondary references at best.

I also recommend looking on all rubber parts with suspicion because rubber does not age gracefully. Check the date codes on your tires and replace them if they are over 5 years old no matter how good they look & feel (old rubber simply cannot flow around the irregularities in the asphalt well enough to grip, especially if it is cool or wet). The original rubber brake lines should be replaced every 2 or 3 fluid changes (= 5 or 6 years) so if your bike still has them I recommend shopping for modern stainless braided ones (they last practically forever and double the life of the fluid). And don't forget things like the rad hoses and the boot between the engine and swingarm (they can crack on the bottom where you don't see it).

The best advice anyone can give you about customizing any vehicle is to get it safe & reliable in more or less original condition and use it for a while before you start making any changes so that it can tell you what changes are needed to make it do what you want/need better and then putting your time/effort/money into those instead of wasting those precious resources on making changes based on style or on copying what someone else (who may or may not really understand how the changes affect the way it works) has done.
Modifications that actually make it work better not only are more likely to produce something you actually want to keep and use but also something that doesn't look like every other copycat "custom" around. Mods for the sake of style, on the other hand, often result in a piece of expensive yard art that you can't stand sitting on for more than a few minutes and might even be dangerous.

The solenoid Mike linked to is the type that was used in the GL500, CX/GL650 and '82 only CX500 (and many other models starting in the early '80s).

A bit of history:

• All CX500 models up to the 1981 model year came with Capacitor Discharge Ignition (CDI) that is powered & triggered by extra windings on the alternator's stator (since this took up space in stator there was less room for the charging system windings so the stator's output is only 150W).
• In 1981 they replaced the CX500 and CX500D with the GL500 and gave the GL Transistor Amplifier Ignition (TI), which is powered by the charging system (since the entire stator is available for charging windings the stator for bikes with TI is rated for 252W).

They kept the CX500C for '81 but it was more or less the same as the 1980 model (including the CDI).

• In '82 they introduced the CX500E (not imported to North America) and upgraded the CX500C to TI, doing away with this particular version of CDI for good.
• In '83 they replaced all CX500 and GL500 with the CX650 and GL650 models, also with TI.

Early CX500 models came with a glass tube main fuse in an inline holder. At some point during production that was changed tothe kind of main fuse holder shown in your pic & the page Cats linked to for all CX500 models except the CX500E. The open link fuse (commonly referred to as a "dogbone fuse") and the holder look like this (note the separate cover that you have to remove to get at the fuse).


GL500, GL650, CX650 and CX500E (& many other Honda models) came with the same dogbone main fuse under a door on the side of the solenoid like this

Note that the dogbone fuses become brittle after long exposure to the air and often crumble to dust when touched so we usually recommend replacing it with a modern blade type fuse in an inline fuse holder.

Some years later (long after the CX/GL500/650 family was discontinued) Honda changed to a similar solenoid but with a modern blade type fuse on top instead of the dogbone fuse under the door like the one Mike linked to on Amazon.

Either of the solenoids with the fuse holder attached could be made to work on your bike but it is not a simple plug & play change. This is why I recommended looking at the wiring drawings for the CX500 and the GL500 to see what needs to be done.