As an attorney who has dabbled a bit in family law I was also taken aback by the statement of a stay at home single parent.
In CA, how long you get alimony depends on how long you've been married and usually adds up to half the marriage years (20 years married, 10 years of alimony). If you are married less than 10, then you are expected to get your butt to work and darn quick because the alimony won't last long at all! And even in longer marriages, you are expected to get to work eventually. And all that does anyway is lower the amount of alimony you get.
So there are any number of scenarios that could explain this. Just child support would not be enough to live on unless she's got lots of kids and the ex makes lots of money to qualify to pay lots of child support.
I'm feeling a little suspicious, but without more information it's hard to come to a conclusion. "Single parent" could mean anything - widowed, divorced, never married. But NOT WORKING and single with children sounds a little strange to me. She has to be getting money from somewhere and all I can think of is lots of money saved on the side, independently wealthy, or she was married a good long time to a rich guy who's still paying her alimony (and child support) and she lives in a state where she's not required to be financially independent after divorce (which seems odd).
Regardless of her financial situation, many of the suggestions people have made for dealing with Mr. Frugal make sense. This is definitely a communication issue and she needs to communicate with him and then they negotiate the situation how it works best for them. That's the only real answer any of us can give and it's the best answer.