Just got a 1999 Bonneville last week and it has the same issue that my previous 1995 had. The armrest release button looks like it got pushed too far in.
What has actually happened is that it seems that the top of the button and latch assembly has come loose from the underside of the top of the armrest.
This is the second Bonneville that I have had and the second with the same exact problem which leads me to believe it is a common failure.

Has anyone come up with an easy fix for it?
As of now I have it closed, but it would be nice to have it functioning.
I would really appreciate it if you could share.

I had the same problem and I removed the plastic piece that attaches to the bottom of the lid. Now there is only the molded plastic where the top part of the latch used to be. If I push it down and a little toward the rear it stays closed. I don't even think about it anymore unless I accidentally bump it when dragging something acroos from the front passenger seat and it pops open.

That would be a broken latch or assembly. In lieu of getting another one, I've seen people cut the spring.

Same thing happened to mine. At the same time, the small rod holding in the cup holder fell into the hole, so I had no cupholder or armrest. I eventually put the cupholder in place and then put 1 strip of velcro across the thing, and another on the bottom of the lid. Its worked pretty well, and has held the armrest shut. That is, until it got really hot and the glue holding it on melted lol