If you have the time and want to do a bit of hiking and see some gardens, then Nokonoshima awaits. It's in Hakata Bay and about a 10 minute ferry ride (about 5km, or 3 miles), or less by speedboat from the Meinohama Port. The island also offers some camping as well as bicycle rentals by the hour or for the day. Another great place is the Nokonoshima Observatory where you can view the whole island and bay from an inland peak 195m tall for free.
Aside from the spectacular views of Hakata Bay, there is a huge variety of seasonal flowers to be enjoyed, particularly the nanohana (rape blossoms), cherry blossoms, azaleas, and hydrangea in the spring and the cosmos in the autumn. A popular place on the island is the Nokonoshima Island Park (entrance fee 1000 yen), which also has several games for children. The return boats shut down service before 11PM though, so be sure to find out when the last boat leaves and don't be late! You can walk to the ferry port from the Meinohama subway station in about 20 minutes, or you can get a bus from the station, or from the Tenjin Bus Center.
Here are some guides on the islands and ferry schedules (in .pdf):
