How to remedy possible ECU on my mini Cooper convertible 2006?

FIRST OF ALL you cannot SUCCESSFULLY use a BATTERY from AUTO ZONE or other CHEAP batteries to START WITH! You MUST use only EURO or ASIAN batteries that are of HIGHER quality and PREICISION! POWER WASH your ENGINE with GUNK ORIGINAL ORANGE engine cleaner at a DIY CAR WASH and clean the SPACE between the RADIATOR and the AC condenser where LEAVES< TWIGS< BUGS< and ANIMAL nests and FECES have collected for YEARS! This will improve ENGINE BREATHING and keeping things COOL! NEVER EVER use VALVOLINE but only REAL EUROPEAN full synthetic oil with the LONG LIFE, LL-02 RATINGS! USE PENNZOIL ULTRA EURO OIL in the 0W-40 rating as it protects to 35 BELOW ZERO >>> A FAILNG BATTERY can affect the RELAYS and ALSO proper VOLTAGES! SO your choice of a BATTERY is CRITICAL! AUTO ZONE batteries are JUNK anyways! >>>THE ECU means ELECTRONIC CONTROL UNIT and applies to the ENGINE running ONLY! There May be other COMPUTERS involved in this but USUALLY it is only the RELAYS! A CLEAN ENGINE is ALSO a great place to START! REMOVING all traces of DRIPS or LEAKS or  LEAVES and ROAD DEBRIS! CLEAN inside the engine and OUTSIDE the engine is the GOAL to LONGER ENGINE LIFE! >>>a BATTERY may have INTERNAL SHORTS in it! and many times even a GOOD TEST will NOT reveal this! TRY A DIFFERENT BATTERY first and be SURE all the CABLES are SHOWROOM CLEAN! 

The Mini Cooper is an electrical nightmare.  Take your car to very good car electrician before spending a fortune on a new ECU.

Why do you suspect the ECU? 

It has nothing to do with what you have reported. It’s rather like if you sprained your ankle and immediately blamed the inevitable pain and loss of function upon brain cancer.

Water has got into your car electrical system, probably due to a leaky roof; if that’s the original roof then it’s years past due for replacement.

The problem is likely two or more faults. The roof operation is controlled by relays and microswitches, plus the roof operation switch. All of those can and do fail, and BMW aren’t renowned for the longevity of the electrical parts they fit. Failure is typically due simply to small, cheap, parts which wear out or corrode (due to getting wet!). Like most electric roof convertibles, some of the cheapest bits to buy can be the most labour intensive to replace.

The intermittent dial lights will typically be due to water ingress into the multiplug connector behind the dial unit, or due to water inside one or more fuse boxes (IIRC, your car has one fusebox inside the car up by one of the front footwells, and one inside the engine bay: I may be wrong). 

The roof control relay will also typically sit inside one of those fuseboxes. So if water has got into one of the fuseboxes then it can affect the dial lights AND the roof control relay. But with that roof, don’t be surprised to find additional faults working together.

The ECU is very robust and reliable and is very well sealed against ingress of moisture. If it had a fault then the car would either fail to start or would run very roughly. It would also display an appropriate warning light on the instrument binnacle.

I strongly suggest that you join one or more of the many BMW Mini online forums. Such as this one:-

https://www.mini2.com/threads/mini-cooper-ecu-loca...

EDIT: As a *TEMPORARY* fix, try spraying WD40 into any wet electrical parts. Lots of people totally forget that the principal design purpose of WD40 is as a Water Dispersant. 

Also like most modern cars, the brake light circuit probably remains active when the ignition is turned off. So if your brake lights flash intermittently after removing the key that suggests the brake light switch is failing: not unusual on a Mini that old. You may well find that the brake lights keep flashing on and off without you even being aware when you’re driving. Get someone to follow you as you drive around the block a few times to check.