Monads in a bicategory $\mathscr B$ correspond to lax functors $* → \mathscr B$, so one expects morphisms of monads should correspond to nautral transformations between them.
A natural transformation between two lax functors $F, G : \mathscr B → \mathscr B'$ is given by a family $φ_B : FB → GB$ of 1-morphisms in $\mathscr B'$ and a 2-cell $φ_f$ replacing the usual commutative square. Requiring $φ_f$'s to be invertible is obviously too restrictive, but then one must choose a direction: $φ_f : Gf ∘ φ_A ⇒ φ_B ∘ Ff$ (what nlab calls lax natural transformation) or $φ_f : φ_B ∘ Ff ⇒ Gf ∘ φ_A$ (oplax).
Spelling this all out for the familiar case of monads $(\mathscr C, T)$ and $(\mathscr D, S)$ in $\mathrm{Cat}$, we have a functor $F = φ_* : \mathscr C → \mathscr D$ and a natural transformation $φ = φ_\mathrm{id} : SF ⇒ FT$ in the lax, or $FT ⇒ SF$ in the oplax case, and everyone seems to agree that lax is the way to go. So I guess my question is: is the oplax case uninteresting and is there a reason it's uninteresting, or am I misunderstanding something? (My understanding of 2-categories is very basic and superficial so I'll admit that the second option is a strong contender)
Note that when $F = \mathrm{Id}$, which is the case I'm somewhat familiar with for $\mathscr C = \mathrm{Ab}$ (where cocontinuous monads correspond to rings) and $\mathscr C = \mathrm{Set}$ (finitary monads = algebraic theories), it doesn't matter which direction we choose because they're dual (although the oplax one which collapses to $\operatorname{Hom}(T, S) ⊆ \operatorname{Nat}(T, S)$ certainly looks like the more sensible choice, and it's exactly what you get if you take monads on $\mathscr C$ to be monoids in $\mathrm{End}(\mathscr C)$.
Finally, for $\mathscr B = \mathrm{Rel}$ monads are preorders, and unless I miscalculated somewhere, the oplax case is the one that gives the expected notion of a morphism and this is actually what made me think about the direction of that 2-morphism.