I'm curious, what is the reason for your daughter's car needing a new engine?  spun bearing, rod knock, seized, over heated, etc...?  
I have an 08 and an 09.  Both are R56's and I've done just about everything on these cars.  But I don't need to replace any engines...

I do have some serious recommendations:

Cravenspeed dipstick upgrade - worth every penny.
Oil and filter changes every 3-5000 miles max
Check oil and coolant level every time you gas up.

THANK You all for the input! we are getting a second opinion and will look up trusted MINI shops in our area! Much appreciated

I would definitely get a second opinion.  See if you can find a MINI club in your area and aske them for a referral to a trusted MINI shop.  Here is a listing for clubs on our site - may not be up to date but a place to start.

I would be curious to hear 'why' your MINI needs a new engine....

You would be best to take the car to an independent MINI mechanic. You can find them over through some of the MINI (all caps) forums. This here is predominantly a classic mini (lower case) forum.

It indeed sounds to me that your daughter is being taken advantage of, but not out of malice and forethought, just like they offered you a price that was a "we don't want to do the job but if you're dumb enough to pay it we'll do it" sorta thing.

The 2008 has many common problems, the most predominant is that it consumes oil at a pretty significant rate. Sometimes its a pcv valve diaphragm sometimes it is valve guides that move up and down in the head casting. You really need to check the oil at each fill up. Oh, and the dipstick needs to be modified or you need an aftermarket dipstick or else you'll get false readings. The stock dipstick will pull up some oil from the sump into the dipstick tube. So when you go to wipe it off, stick it back in, and check the level it'll show that it's full. You need to leave the dipstick out for a good 10-15 minutes and let the oil run back down the dipstick tube and then stick it back in and pull it out to check. only then will you get a better idea of the level.

The 2008 we had used 2 qts ever 3 fill ups, or about 3/4 qt every 300ish miles and it had under 100K miles on the clock.

try northamericanmotoring.com or mini2.com  website forums for more reading and specific advice on where to go based on your location.

LCMini..tons of the stories on bad engines with the BMW minis.. just buy good used one off www.car-parts.com   should be able to get one in the $2-3k range..  later bc

My daughter recently purchased a 2008 2dr hatchback 1.6L I4 F DOHC 16V Gasoline front wheel drive Mini Cooper.

It had 55K miles on it and the CarFax all showed it was in good shape.  It had recently been serviced by the dealership and had the oil changed.

Fast forward 5 months and the dealership is now saying it needs a new engine and that it has several oil leaks.  My daughter has had the oil changed twice in this time frame and has taken very good car of this car.  

Has anyone had a similar experience and if so, what are the solutions? Dealer ship quoted $14K for a new engine? Seriously??

I think they are taking advantage of my daughter, but not sure what to do now.