I have two thoughts on this:
1. I agree with the others about her technique. I'm not saying this in a bad way, but moms tend to have the hover/technique built into their mom-brains. When I worked at Ritz Camera (please don't throw anything) I used to get people trying to return cameras constantly because they said they were slow or that they missed all the shots or the images were blurry. If I took a moment to show them that they can hold the shutter button down halfway while their kid is at bat and then finish the shutter press right as they were swinging, they'd get a lot more keepers. Boom. Done. I think if you teach her this, she'll get a higher keeper rate and probably find that the little Sony is fine.
2. Right off the bat, people are mentioning Ricoh (not a good camera for people who want snapshots of their kids), Canon S95, G10, etc. If she's going to spend $400-500 on a high-end point and shoot, she'd be better off buying an inexpensive DSLR (Canon has a refurbished XS kit for $400). The DSLR will outperform any point and shoot (especially an older one) in terms of speed of focus/shutter lag. I think the whole pre-focus or setting it to f/8 and 5m is not what she will be looking for and would probably just confuse her, especially if her camera doesn't have the capability.
I'd say teach her to half-press BEFORE the moment and finish the press as the moment happens. Or sell her on an inexpensive DSLR with kit lens.