[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 $d$. (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.)

[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 $d$. (Use sections of a degree-$d$ divisor to embed in ${\bf P}^{d-g}$, and project down to a generic ${\bf P}^3$.) But ${\cal M}_g$ is of general type once $g \geq 24$ (Harris-Mumford-Eisenbud), so  $H_{g,d}$ cannot be rational or even unirational for such $g$.

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.)

[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.)

[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 $d$. (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.)