This is the 13143rd most frequent Russian word.
меньшую
"Меньшую" means "smaller" or "lesser" (feminine form, accusative case).
The word 'меньшую' (a form of 'меньший', meaning 'smaller') is used here as an adjective in the accusative case, modifying the noun 'box' (коробку), indicating the object being chosen.
Я выбрал меньшую коробку для хранения.
I chose the smaller box for storage.
'меньшую' describes the noun 'portion' (порцию) and shows selection of a smaller amount, used in the accusative case as it's the direct object.
Чтобы похудеть, я ем меньшую порцию пищи.
To lose weight, I eat a smaller portion of food.
'меньшую' modifies 'book' (книгу) and is again in the accusative case due to being the object of the verb 'prefer' (предпочитаю).