Why not look into Bayonne ?
http://www.nytimes.com/2014/08/24/re...e-nj.html?_r=0
I've been living here for 2 years now, and I don't want to move anymore...
I rented here for 1 year before I bought my cozy house.
As a young professional work in downtown Manhattan, 45 minutes door to door daily commute is very reasonable for me. (my work location is located close to the WTC path station. )
Most of my friends & young colleagues lives in fancy areas like Water Front, NJ or even in downtown Manhattan and pay insane amount of rent (like 2x or 3x more than my monthly mortgage payment). They like to troll me for living in a "un-cool blue-collar town" like Bayonne.. But then I troll them back with "Hey, you guys are still RENTING while I OWN my house as well as a 2014 Mercedes which I park it in my own garage with the fact that you guys still use ZIPCAR !!" (LOL)
I used to live near the Grove St path station, where I had a 600sqft 1bd apartment for nearly 2k a month. The life there was okay, yet apart from being close to the Path Station, which means faster commute, I didn't really get anything partially worth for the amount of extra money I paid. Well, nice restaurant and bars if you want to count them as well.
As for the OP, I suggest you look into rentals at E45st ~ E20St on Avenue E, Bayonne.
You'll find lots of place within your budget as well as a much safer and family-friendly community than most parts of Jersey City, not to mention the easy commute to Manhattan.