[expanding on my comments above]


$H_{g,d}$ is rational for some but not all $(g,d)$.

A rational example is $H_{1,4}$: a quartic elliptic curve in
${\bf P}^3$ is the complete intersection of two quadrics, so
$H_{1,4}$ is just the Grassmanian of $2$-dimensional subspaces of the
$10$-dimensional space of quadrics.  Similarly for other cases where
the curve must be a smooth complete intersection; the simplest examples are
$(g,d) = (0,1)$ and $(0,2)$ if you allow curves that do not span ${\bf P}^3$.

On the other hand, given $g$, for $d$ large enough the Hilbert scheme
$H_{g,d}$ dominates the moduli space ${\cal M}_g$ of genus-$g$ curves,
because every such curve can be embedded in ${\bf P}^3$ with degree $3$.
(Use sections of a degree-$d$ divisor to embed in ${\bf P}^{d-g}$,
and project down to a generic ${\bf P}^3$.)  Once $g \geq 24$
${\cal M}_g$ is of general type (Harris-Mumford-Eisenbud), so
$H_{g,d}$ cannot be rational or even unirational.

The question of characterizing all $(g,d)$ for which $H_{g,d}$ is rational,
or even unirational, might be quite hard.  (I gather that "unirational"
can be more accessible; e.g. the Hilbert scheme $H_{0,3}$ of rational
normal cubics is certainly unirational but it's not obvious to me that
it is actually rational as I rashly claimed in my first comment.)