According to Jeff Wisnia :
Methinks that problem is going to be a bitch to find and correct, and
hiring professional help to do that will prolly cost a lot more than if
the OP just purchases a new window AC for his mom.

It might just need a good cleaning.

If it was me, and there was an equipment grounding conductor in the AC's
power cord, I'd just verify that the receptical was properly grounded
and then chop off that GFCI plug and replace it with a standard grounded
plug. But that's me, and I'm not suggesting the OP does it, 'cause his
skills are as yet undefined.

That'd be a very last resort. I don't like mysterious ground faults,
and if the ground is the slightest bit dubious (not able to sink well
over 20A), this is a disaster waiting to happen if there's no GFCI.

If I was convinced the GFCI was going bad, I'd hack off the GFCI
plug and replace it with a regular plug. _Then_ either put a GFCI
outlet in place of the existing receptacle, or, get a (short!)
large-gauge wire GFCI extension cord or GFCI'd power bar and plug
the A/C into that.
--
Chris Lewis, Una confibula non set est
It's not just anyone who gets a Starship Cruiser class named after them.