Dear Ted, First of all, thanks from the bottom of my heart for your service to our country, and God keep you and your company safe.
I'm not legally allowed to give my opinion as to whether a neighborhood is safe or not, but check out AZcentral.com and go to crime maps and you can see how neighborhoods stack up. Be sure to check more than one. (When you first see 200 crimes in a year, it looks like a lot until you realize many zip codes have that many. Check out the type of crime too; its right there on the site.) Augusta ranch has some shopping; if you go to google maps and do the satellite view, you can see where the shopping areas and parks are in other places as well. If your mom needs a place within walking distance or using public transit, the specific location within a neighborhood becomes important too.
Definitely there is good value to be had in Gilbert, Chandler and Mesa, and a number of places that could be great for your mother.
In terms of weathering the market, that is hard to call but I'd advise purchasing a home in good condition, in a neighborhood that is well-maintained. The good news is foreclosed and short sale properties under $200k are selling quickly, auguring well in the long run for the overall market. As those properties disappear, regular resales will again become the norm, and at that point we'll have a genuine recovery.
Is your mother able to go look at properties? I definitely advise against a purchase without seeing first. Your price range is key here; if you'd like some more assistance please email me at email@example.com.