Hey Sarah,
I have been investing in Chicago real estate for 18 years and I'm very familiar with the south side ( I grew up there).
Some of the gentrification areas are around U of C in Hyde Park such as North Kenwood, Oakland, Woodlawn and Douglas. Bronzeville, just south of the Loop continues to have good deals. The Morgan Park/Beverly areas and Ashburn are nice for investors. South side Chicago have many neighborhoods that are block by block situations. Meaning one block may be well manicured and look like a sprawling suburb and a couple blocks over the houses are total eye sores. In this case the eye sores have brought down the price of some very nice properties which provides great opportunities for the buy/hold investor.
In regards to eviction, screening should be your first and most efficient line of defense. Cook county can be very lenient to tenants but if you are a proactive landlord you will be fine.
Nice Investing