Yes, I have a TON of thoughts... some of them pertain to that computer....
The HP slims have a known mobo issue. Take that to the bank, my friend. When you talk to customer care, they will play dumb, or they are dumb to the situation.
In fact, the last few desktops, slims, and laptops that HP has produced all have mobo issues. Some even have a recall, and the documentation for that is buried in their web site. It is a design fault in the pcb's that has never been resolved. The Slim desktops and dv*000 series laptops and tx*000 tablets fail the most.
If the computer is less than 24 months old, and it is on "the list," then HP has to repair/replace it. If they will not work with her on that part of the issue, then I strongly suggest NOT to pursue the repair anymore. Any motherboard used as replacement will have the same underlying issue and will meet the same fate, and the HP replacements are usually underclocked and have features disabled. I have seen this many, many times.
Sad thing is, almost all off the shelf desktops are really cheaply built, more so then they were just a few years ago. Her next pc needs to have a minimum 3yr warranty attached to it.
Sorry for the bad news, buddy. Screw HP... right in the pooper.