What have you tried? Have you used Object spy to figure out the object and it's methods? You can try recording selecting the item and see what TestComplete finds. Once you do so, you can make whatever modifications you want.
Try to catch the item you want to select with object spy with point and fix and choosing the specified item and press Shift+Ctrl+A in this way you can see if the class of the dropdown is recognized.
I had the same problem with selecting the dropdown items on desktop application build in Delphi, after adding the class of the dropdown Tools->Current Project Properties->Open Aplication->Text Recognition all items from the dropdown were recognized.