The later ones had a BMW diesel unit. The earlier PSA units were OK, but were very maintenance fussy and didn't tolerate any kindo of neglect or abuse. Of course, prats bought them and then upset them by using them for short journeys, and after 3 years most ended up at Fred in a Shed for servicing where they rarely got fed the diet of expensive low SAPS oil they required, so a lot of it isn't down to the engine itself per se but the way they were treated.
The BMW-PSA "Prince" engine was troublesome, but rarely serious. I'd have to check but by 2018 that engine had made way for the 1.5 BMW unit (not 1750) on the Cooper which is better, albeit not foible free (what engines are.)
Unless they're prepared to have it professionally inspected prior to purchase and take it to somewhere reputable for service (and not just then local independent on an industrial estate) I'd steer them elsewhere. They're well made vehicles and can be very long lived, but that engine needs attention from people that know them and who are in a position to apply then periodic software updates BMW release, which keep the engine sweet and the sat nav up to date.
Check with the BMW dealer, because annual service plans are dirt cheap, often as little as £15 a month and they include MOTs as well.
In short, it was a premium product when new and 7 or 8 years on it needs the proper care and attention it deserves else they might just as well buy a Fiesta.
