Cinaralti Cay Bahcesi in Cengelkoy is right by the Bosphorus strait. It is a great place to mingle with locals as well. It's on the Anatolian side. It still keeps its tea garden spirit, not turning into a cafe.
One of my favorites is the one by Rumeli Hisari on the European side. It's still a popular hang out among university students, and locals of Rumeli Hisari neighborhood. But do not expect an actual garden, it's just a large terrace infront of a tea shop. Another famous one European side of Istanbul is the one in Eyup Pierre Loti. It's overlooking the Halic lagoon.
Similarly tea houses at Camlica Tepesi and Hidiv Kasri have great views. Both on the Anatolia side of Istanbul.. Hidiv Kasri and surroundings also offers a piece of Ottoman history as well.