For an ordinary functor $F\colon \mathcal{C} \to \mathcal{D}$ of categories, there is a construction $\operatorname{im} F$, the *image* of $F$, which is again a category, and $F$ factors through that image.

Is there anything vaguely like the image of a strong monoidal functor, which should be monoidal again?

What certainly doesn't work straightforwardly is just taking the image of the underlying functor and somehow putting a monoidal structure on it. Given $X, Y\colon \mathcal{C}$, how would you define $FX \otimes FY$? If you take the tensor product in $\mathcal{D}$, the result might not be in the image of $F$, but neither can you define it to be $F(X \otimes Y)$, since $F$ might no be injective on objects (so you can't find out a unique object to start with, and there is no canonical way to choose).

I'm utterly surprised I've never encountered such a construction. The only thing I've come across is for the case of fusion categories. There, the full subcategory spanned by summands of objects in the image of $F$ can be defined.

The only thing I can come up with for plain monoidal categories is the category spanned by the image of the underlying functor and all isomorphic objects. But this doesn't have good properties if I want to extend structures on $\mathcal{C}$ to $\mathcal{D}$. For example, if I have a braiding $c_{X,Y}\colon X \otimes Y \to Y \otimes X$ on $\mathcal{C}$, I can transport it onto objects of the form $FX$, but not onto all objects isomorphic to an $FX$, since I don't know which isomorphism to transport it along.