One thing. Did they try removing the SD card? SIM?
http://www.gsmarena.com/samsung_galaxy_mini_s5570-3725.php notes this is from 2011 so it's well past it's prime.
PS. http://www.ebay.co.uk/itm/Samsung-Galaxy-Mini-GT-S5570I-Steel-Grey-Unlocked-Smartphone-Faulty-/171669934459 finds other failed units.
Simple problem,complicated solution. I have been to various forums for the answer on how to get to recovery mode but none seem to help. All the various button combinations....press power/home/volume key etc don't want to work. All they do is take me to download,Ramdump etc.
Original problem is flashing Samsung icon