That sounds suspiciously like a blown clutch diode. It is a 3 blade component, looking rather like the two blade fuses on the ST1300. I think its a bit larger on the 1100. It plugs into a connector in the main wiring harness, and is often covered over with the tape that is wrapped around the harness. So to find it you have to feel for the bump. Someone with access to an ST1100 will be able to tell you exactly where it is, but if I remember correctly, it is in the thick loom that passes down the frame under the left hand side, under the dummy tank cover. Shown as Part#1 on this diagram for a 2001 ST1100A, with the corresponding socket shown in the loom.
I cannot say that it is definitely that, but it sounds like a good candidate from my experience.
If you decide that it is that, then before you replace it, consider that something caused it to blow. In my case it was when I was fiddling with the starter relay - a dodgy connection probably put 12v down the wrong line. Diodes / rectifiers don't like that.
Basically, the neutral light is connected to a number of things. From memory these are the neutral position sensor, the clutch lever and the side stand. The diode prevents the voltage from one of these passing to the other. You'd need to look at the circuit diagram to work out how it connects. I guess that you could fashion your own with two diodes connected pointing towards each other and plugging them into the connector. You'd need to work out what current rating is required. I doubt that it is much - its only the neutral light.
When mine blew, it would come on every time I pulled the clutch lever in when riding. The side stand switch also plays a part, but can't remember exactly how that showed up.
Part # 31700196000 - google it to see what it looks like, but please check the part number for yourself.
to you Bush.