Darien, Downers Grove (West 'Burbs) aren't too bad, but DuPage county in general is going to be a little high.
In the city proper (not downtown), the rent and housing costs are less expensive, but other things are very high. For example, car insurance will be (at least) double in the city (over far-West 'burbs). The City makes you buy a "City Sticker" annually; when I left it was up to $75.00 a year. So, from an ongoing cost perspective, the city looks cheaper, but may end up costing more.
If you can stay out of Cook County (like to DuPage, Will, Kane, Lake) you'll ultimately be way ahead of the game, but houses and apartment will tend to cost more (more in DuPage and Lake ... Will and Kane are WAY West and South - one hour + train ride to downtown).
Within Cook County, some of the closer 'burbs aren't too bad, can be reasonable rent/house-wise, and don't have you bending over tax-wise. Places like Shiller park, Stone Park, LaGrange, Countryside, and Bensenville are pretty reasonable trade-offs. Metra is the suburban transit system -
Metra - The web site If she's gonna be commuting by rail, check out the rail lines and travel times and see where the pain threshold begins....
Parking downtown (montlhly rate) is over two hundred dollars a month in the loop, slightly less as you get further away.
What it boils down to (IMHO) is rent/house (+ taxes) and car insurance. A female friend of mine lives in Forrest Park, she's Fourty-something, no accidents (ever), no tickets (ever), drives a 95 Chevy Blazer. Her insurance is ~$2000+ / year with Allstate (full coverage). Forrest Park is next to Oak Park, which is next to the City of Chicago (Forrest Park is ~1 mile from the city boundry).
I live in Wheeling, fourty-something, no tickets (~15 years or so), no accidents, drive a '00 Jeep. My insurance just went up to ~800.00 a year (full coverage). Wheeling is 15-20 miles from the City of Chicago boundry.
Depending on how tolerant she is for train rides, she may even want to look someplace like Kenosha Wisconsin. Cost of living is much less, and the train ride is still ~one hour.
It might be a good idea to take a long weekend trip to the Chicago area and scout it out ... see some sites.
Does she already have a job here? What kind of work does she do? Chances ae she can get a similar job out in the 'burbs.
Good Luck. If you need some specific information, let me know; I'll see what I can gather.
Scott