Best family hotel Bayonne NJ
There isn't much in Bayonne, especially as far as hotels. I also have no first hand knowledge of hotels in the area, as we usually spend the night in Manhattan before a cruise out of Bayonne because my In-laws let my children stay with them in CT and then will meet us at the port.
I would recommend looking at hotels near Newark Airport. There should be plenty of options available depending on if you want a regular room or a suite. Hotels in that area are still about a 20 minute ride to the port, but this will give you the best options and the widest price range so there should be something within your budget!