Let $\mbox{Inv}(\mathbb P^n)$ be the
subgroup of $\mbox{Bir}(\mathbb{P}^n)$ generated by involutions and
$\mbox{PGL}(n+1, \mathbb C)$.

If $n\ge 3$ then any generating set of $\mbox{Inv}(\mathbb P^n)$
contains uncountably many involutions. More precisely
for any $d >1$ there are
uncountably many
involutions of degree $\ge d$ in any generating set of $\mbox{Inv}(\mathbb P^n)$.

This is a direct consequece of the proof of Hudson-Pan Theorem which asserts that $\mbox{Bir}(\mathbb{P}^n)$, $n\ge 3$, needs uncountably many generators of degree greater than $d$ for any $d>1$.

Let me briefly review Pan's proof adapting it to prove the remark above. It is achivied in two steps:

for each hypersurface $H$ of $\mathbb P^{n-1}$ there is a birational transformation of $\mathbb P^n$ which contracts a cone over $H$;

a hypersurface contracted by a product of birational transformations $f_1 \circ f_2 \circ \cdots \circ f_k$ is birational to an hypersurface contracted by one of the birational transformations $f_1, \ldots, f_k$.

The proof of 2 is straight-forward. To prove 1, fix $p \in \mathbb P^n$ and consider the subgroup $\mbox{St}_p(\mathbb P^n) \subset \mbox{Bir}(\mathbb P^n)$ which sends
lines through $p$ to lines through $p$. One can show that $\mbox{St}_p(\mathbb P^n)$
fits into the split exact sequence
$$
1 \to \mbox{PGL}(2,\mathbb C(\mathbb P^{n-1})) \to \mbox{St}_p(\mathbb P^n) \to \mbox{Bir}(\mathbb P^{n-1}) \to 1
$$
where the rightmost map is defined by the action on the space of lines through $p$, and the leftmost is defined by the action on the fibers of the $\mathbb P^1$-bundle obtained from $\mathbb P^n$ after blowing up $p$.

Now given $h \in \mathbb C(x_0, \ldots, x_{n-1})$, we can consider the element of
$\mbox{PGL}(2,\mathbb C(\mathbb P^{n-1}))$ defined by
$$
(s:t) \mapsto (t:h\cdot s) \, .
$$
Clearly, this defines a birational involution of $\mathbb P^{n-1} \times \mathbb P^1$
which contracts the divisor associated to $h$. Of course, we can realize it as an element
of $\mbox{St}_p(\mathbb P^n)$ inducing the identity on $\mbox{Bir}(\mathbb P^{n-1})$ and
contracting a cone over $H=h^{-1}(0)$ with vertex at $p$.

Notice that we can choose uncountably many $H$ in uncountably many distinct birational
equivalence classes. Putting this observation together with 1 and 2 allow us to conclude.

You are probably
aware of it, but there is classification of birational involutions of $\mathbb P^2$ by
Bayle-Beauville which completes and clarifies previous works by Bertini and others.
Composing the involutions of $\mathbb P^2$ with elements of $PGL(2,\mathbb C(\mathbb P^2))$
gives many examples of birational involutions of $\mathbb P^3$ which are also in $\mbox{St}_p(\mathbb P^3)$.

For more on $\mbox{St}_p(\mathbb P^n)$ see this other paper of Pan.