Look into Brooklyn or Queens or far uptown Manhattan, or the Bronx - it's far easier to get there than it is from Jersey City. Tons of people commute to the whole First Ave medical area (1st Ave from the VA at 23rd up to around 35th) that way.
In my opinion, Queens is a better value these days - if you can make two or three friends to have as roommates, you can get a really great 3BR apartment in Queens. With one roommate, it becomes slightly harder, but still doable to find a great place that fits a student budget.