From what I can tell this is just one of the quirks with the early minis.
Mine comes on from time to time, so I ended up spending £100 on a Sealy diagnostic tool so I can reset the light.
When it comes on, I unplug the yellow and blue connectors, give them a squirt of contact cleaner - connect and disconnect them a couple of times to make sure the connectors are clean and then reset the light.
Typically this is fine for a few months or until someone climbs in the back and disturbs the connection!
Don't let it spoil the fun as they are still great cars - I would suggest save up for a cheap diagnostic tool so you can reset it yourself. At £10-£20 a reset it will pay for itself quite quickly