To be honest, I wouldn't want to live near JFK between 49th and 56th. There isn't much there except the park and you are far from the light rail. There was a fair bit of crime along that stretch during the summer.

My Wife and I made the plunge from Downtown JC to Bayonne 2 years ago. Its not the worst place to live and it's not the best place. There are things we really like:

The parks
more square footage,
owning,
close to the city,
The beer is cheap,
Close to shopping (Stop and Shop, Loews, Houlihans, Walmart, Michaels)
Decent public schools

I've spent 15 years living in Jersey City and watched it evolve. I was reluctant to leave. You can get some really good deals on a house. We bought a 4 bedroom single family with basement and backyard for a little more than what we were paying in rent. Bayonne has a lot of potential but there are things that bother me:

the light rail is super slow and doesn't run all night
Little choice for take out (mostly chinese and italian)
The beer is cheap (every bar is an old man bar)
little or no art/music scene
Feral cats
You need a car to get around
People in Bayone blame Jersey City for everything.

All in all, I'm happy with the choice, but there are times I miss living downtown. I'd murder a curry right now.