My 2005 Mini cooper converttible engine is

I Would Start With Repairing The Radiator First Once It Does not Leak Your Problem Will Probably Be Over

Check that your radiator fan is working if not check fan belt if it has one , if not check fuse . Next suggestion if all these are ok you may have a thermostat fault check the wires to this it may need relacing..

More than likely the cable on the inside of the right mechanism snapped. To repair it, the entire mechanism needs replaced to the tune of upwards of $3000 USD. Sorry to be the bearer of bad news. And even worse is there is no real way to check it without pulling the mechanism apart. I hope you find this information useful. You may wish to stalk the junkyards for one out of a junker. You can probably hook one of those up for about $300.00. You may also wish to look for the lift mechanism. Here is a place that has both, pretty reasonably priced. They could also help you to pinpoint the problem better based on you actual experience.

2005 Mini Cooper Convertible Top Motor

http://www.automotix.net/usedautoparts/2005-mini-cooper-convertible_top_motor-product-9zy26179d.html

Convertible Top Lift, Passenger Side
http://www.automotix.net/usedautoparts/2005-mini-cooper-convertible_top_lift-inventory.html

A 2005 mini cooper s does not have a timing belt. the engine is driven by chain and is probably good food for 200000 miles or untill they start rattling.
You will just need to inspect the super charger belt. If there are no cracks in it it will be fine.

In the S model it's in the boot under the floor mat.
In the standard model it's under the bonnet on the left sidr of the car (US Drivers side) You'll see the cables going into the plastic battery box.

Change your Minis Mass Air Sensor this will solve your problem.

There is not a radiator cap on mini coopers it is a closed radiator. The coolant expansion tank is in the back of the engine compartment twards the middle. The fill for the coolant is just on the drivers side of the engine block between the engine and the battery box/ engine air filter.

There is a problem with the thermostat gaskets. Often times they leak and allow cars to overheat. Check under your car and make sure that there is no coolant leaking. MINI has updated the gasket so if you get your car serviced at a mini dealer then your gasket will be updated. There is not a bulletin or recall for this replacement this would be at your cost.

The main cooling issue that is for the 05 cooper is the Thermostat gaskets leak. MINI has updated the part but has not issued a bulletin or recall. Check under your car often and make sure there is not a puddle of coolant. They just updated the part within the last 3 months, so it is still pretty new. The old gasket is a white color and the new one is a semi clear. The only thing that you can do is be on the look out for coolant leaking on the floor in the middle or towards the drivers side of the engine bay, Almost positive that MINI will not allow you to replace a part under warranty unless your thermostat is leaking. Hope that helps answer your question

That is your motor mount leaking, it needs to be replaced