Most of touristic attractions are in European side of Istanbul, however there are also tons of things you can do in Asian side.

Kadikoy: Central of Asian side, nightlife, shopping, history and romance (especially moda district)

Üsküdar: One of the oldest settlement of Istanbul and still you can find lots of historical monuments. Mosques, Churches, Medrese and old houses.

Kuzguncuk: Old wooden houses, church, synagogue and mosque in same area even Synagogue and Mosque sharing same wall.

Çengelköy: Breathtaking Bosphorus view, delicious borek and pastries.

Bostancı: In crossroads of Asian side, a 7/24 living district, shopping, nightlife, posh cafes and restaurants and great experience to walk or cycling near Marmara Sea…

There are more to count, let’s add more districts with comments below…

Thanks to @jon_wertheim for his question.