A note on fonts: below I use mathsf for sentences and mathit for objects. So e.g. $\mathcal{M}\models\mathsf{V=L}$ iff $\mathcal{M}=L^\mathcal{M}$.
Depending on whether you allow parameters in your fixed formula, you're either looking for "$\mathsf{V=HOD}$" or "$\exists xV=\mathsf{HOD}(x)$$\exists xV=\mathsf{HOD}[x]$." For simplicity I'll focus on the parameter-free situation.
Here $\mathsf{HOD}$ is the class of hereditarily ordinal definable sets. (See also this old answer of mine.)
It's easy to show that $\mathsf{V=HOD}$ is equivalent to the existence of a parameter-freely-definable well-ordering of the universe. The subtle feature is that $\mathsf{V=HOD}$ is in fact expressible in the language of set theory! This is a beautiful trick, so below I've spoilered the key one-word hint:
Reflection.
As to the connection with the axiom of constructibility $\mathsf{V=L}$, this is in fact vastly weaker. A good starting point is to understand just how flexible the construction $\mathcal{M}\leadsto\mathit{HOD}^\mathcal{M}$ as compared with $\mathcal{M}\leadsto L^M$; for example, any (countable!) $\mathcal{M}\models\mathsf{ZFC}$ has a forcing extension $\mathcal{M}[G]$ such that $\mathit{HOD}^{\mathcal{M}[G]}=\mathcal{M}$ (this is due to Vopenka if I recall correctly). Of course this doesn't help us build models of $\mathsf{V=HOD}$, but it's a good indicator of how weak that principle is likely to be (e.g. it has no bearing on "arithmetic" issues like $\mathsf{GCH}$, or even - to the best of my knowledge - large cardinal structure as far as we understand currently).