Assume that $V = L$ is true in the ground model $M$, and let $G$ be generic for $\mathbb{P} = Fn(I, 2)$, which is just the set of finite functions from $I$ to $2$. Then, ${\rm HOD}^{\mathbb{R}} = L[\mathbb{R}]$ is true in $M[G]$.


I assume that the proof is just a modification of the proof that ${\rm HOD}$ in $M[G]$ is contained in $L[\mathbb{P}]$ in $M$, but I can't figure out how to do it. 


As in Kunen, ${\rm HOD}^\mathbb{R}$ are the sets hereditarily definable from ordinal and real parameters, and $L[\mathbb{R}]$ is the constructible hierarchy over the reals, i.e., $L[\mathbb{R}]\_0 = \{\mathbb{R}\} \cup {\rm tcl}(\mathbb{R})$, $L[\mathbb{R}]\_{\alpha +1} = {\rm Def}(L[\mathbb{R}]\_\alpha)$, etc., and where ${\rm tcl}(x)$ is the transitive closure of $x$. 


Hopefully the exercise isn't too trivial for mathoverflow. 



Kunen (2011) exercise IV.8.17: $M[G]\vDash HOD^\mathbb{R}\subseteq L[\mathbb{R}]$
This question seems to be a little more subtle than the usual homogeneity arguments. Here is one way that I found to do it.$\newcommand\R{{\mathbb{R}}}\newcommand\HOD{\text{HOD}}\newcommand\Ord{\text{Ord}}$


Allow me to describe the problem like this. We start in $L$ and force with
$\newcommand\P{\mathbb{P}}\P=\text{Add}(\omega,\kappa)$ to add
$\kappa$ many Cohen reals. In the forcing extension $L[G]$, we form
the class $\HOD\_\R^{L[G]}$ of hereditarily ordinal-definable sets.
Meanwhile, $L(\R)$ is the class of sets constructible from reals in $L[G]$,
defined by starting with the real numbers of $L[G]$ and iteratively taking
the definable power set.


**Theorem.** $\HOD\_\R^{L[G]}=L(\R)$.


**Proof.** The inclusion $L(\R)\subseteq\HOD^\R$ is easy to see, as
alluded to in the question, because every real is definable using
itself as a parameter and, as can be seen by induction, every
object in $L(\R)$ is definable in $L(\R)$ using finitely many
ordinal and real parameters, and so every object there is definable
in $L[G]$ using real and ordinal parameters. Indeed, in $L(\R)$
there is a definable surjection $s:\Ord\times\R\to L(\R)$, and this
surjection is definable in $L[G]$.


It remains to see, conversely, that $\HOD^\R\subseteq L(\R)$.
Suppose that $A$ is hereditarily definable in $L[G]$ using ordinal
and real parameters. By $\in$-induction, we may assume that
$A\subseteq L(\R)$. Let $B=\{(\alpha,z)\mid s(\alpha,z)\in A\}$,
which is also definable in $L[G]$, using the definable surjection.
It suffices to show that $B\in L(\R)$.


Since $B\in\HOD\_\R^{L[G]}$, there is a formula $\varphi$ for which
 $$B=\{(\alpha,z)\mid L[G]\models\varphi(\alpha,z,\beta,a)\},$$
for some ordinal parameter $\beta$ and real parameter $a$. Fix
names $\dot B$ and $\dot a$ for which $\dot B\_G=B$ and $\dot
a\_G=a$, and fix a condition $p\_0\in G$ forcing that $\dot B$ is
defined by $\varphi(\cdot,\cdot,\check\beta,\dot a)$.


**Claim.** $(\alpha,z)\in B$ if and only if there is $\dot z$ and
$L$-generic $g$ for a countable complete suborder of $\P$ in $L$, with
$p\_0\in g$ and $\dot a\_g=a$ and $\dot z\_g=z$ and $\exists p\in g$
with $L\models p\Vdash \varphi(\check\alpha,\dot z,\check\beta,\dot a)$.


**Proof.** The forward implication is easy, since we can take $g$
to be a fragment of $G$. Conversely, suppose that we have $g$ and
$\dot z$ as stated. Since $g$ is an $L$-generic real in $L[G]$, we
may extend $g$ to a full $L$-generic filter $H\subset\P$ with
$L[G]=L[H]$. It follows that there is an automorphism of the
forcing $\pi:\P\cong\P$ in $L$ with $\pi[H]=G$. Since
$p\Vdash\varphi(\check\alpha,\dot z,\check\beta,\dot a)$, it
follows that $\pi(p)$ forces $\varphi(\check\alpha,\dot
z^\pi,\check\beta,\dot a^\pi)$, where the superscript $\pi$ means
that we have applied the induced automorphism on names.


Since $\dot a\_G=a=\dot a\_g=\dot a\_H=\dot a^\pi\_{\pi[H]}=\dot
a^\pi\_G$, it follows that there is some condition $q\in G$ forcing
that $\dot a=\dot a^\pi$. Without loss, $q\leq \pi(p)$. So $q$
forces $\varphi(\check\alpha,\dot z^\pi,\check\beta,\dot a^\pi)$.
Since $z=\dot z\_g=\dot z\_H=\dot z^\pi\_{\pi[H]}=\dot z^\pi\_G$, this
implies $(\alpha,z)\in B$, as desired. **QED** (claim)


The claim implies the theorem, since that definition can be carried
out in $L(\R)$, using $a$ and $\beta$ as parameters and the fact that the forcing relation is definable in $L$. **QED**


This argument is a little more complicated than I expected, and I'm not sure if there is a more direct argument.


