An ariticle I am reading states:
Categorically, idioms are strong lax monoidal functors (Mac Lane 1998).
where Mac Lane 1998 is:
Saunders Mac Lane. Categories for the Working Mathematician. Graduate Texts in Mathematics. Springer-Verlag, Berlin, 2nd edition, 1998.
The words "monoidal functor" (pg. 255) and "strong monoidal functor" (pg. 257, but with "strong" meaning something different in Mac Lane, compared to what the article means it to be (see comment by Q. Yuan below) are straightforward to find in the book. However, the word "lax" never appears.
What then is the concept equivalent to "strong lax monoidal functor" in Mac Lane 1998?
I don't have a copy of Mac Lane handy, but from the terminology you've mentioned, I'd guess that the 'lax' means it is the monoidal functor without any descriptor. This means that there are arrows:
$$f : I_D → F I_C$$ $$g_{A,B} : F A \otimes_D FB → F (A \otimes_C B)$$
where $g$ is natural in $A$ and $B$. I'm presuming the 'strong' descriptor (in Mac Lane) means these are isomorphisms. And there might be a 'strict' descriptor where they're equations.
As mentioned in the comments, the 'strong' in the article you're reading means tensorial strength. However, another (possibly related) way of giving a categorical specification of idioms is that they are (lax) closed monoidal functors. This presumes there is an internal hom in each monoidal category (specified for instance as $-\otimes A ⊣ [A,-]$), and that in addition to the above maps, there is:
$$h_{A,B} : F [A, B]_C → [F A, F B]_D$$
natural in $A$ and $B$. This, for instance, makes sense even if $F$ is not an endofunctor.