My Mini One 2003 R50 1.6 litre petrol engine takes specific BMW 5w30 Longlife-04 oil. (I believe the R52 and R53 models take the same oil too.) The oil is 5w30 fully synthetic made to BMW’s exacting standards.
The cheapest I’ve found to buy currently is a GM (Vauxhall/Opel) manufactured one, made to BMW’s specifications. Searching for something like “dexos 2 5w30 gm” on ebay finds them at about £20 for 5 litres with free delivery in UK. (Comparatively, an equivalent from Castrol is about £50 at time of writing.)
Sounds like a small saving, but if your Mini is anything like mine it needs topping up once a month or so, and I do a full oil/filter change every 5k miles as an attempt at longevity. Soon adds up.