I'll endorse the night before option, with the suggestion that it won't matter which airport you use the night before, so you can go with price and schedule convenience.
Also, Priceline and Hotwire hotel availability/prices tend to be better around EWR than JFK (there are more and better hotels around EWR than either JFK or LGA).
Doing the EWR-JFK train transfers could work well on a weekday as long as you go after rush hour. I'd recommend Newark AirTrain to NJ Transit to Long Island Railroad to JFK AirTrain. You're looking at about a $25 fare, on these various trains. I think that there is also an hourly bus that runs between EWR and JFK.