According to GSM Arena, the 2nd generation Moto G uses a
Qualcomm MSM8226 Snapdragon 400
chipset, so it shouldn't be affected by this issue.
(I still reckon my mom needs a new phone though, there are other less severe exploits out there for this non-updated device.)
Yes, an iPhone is better, but if you're on a budget and don't have those security needs, the Moto G seems fine.