5. Given that $P$ is a point on the right branch of the hyperbola $C: \frac{x^{2}}{2}-y^{2}=1$, and $l$ is an asymptote of the hyperbola $C$. The projection of $P$ onto $l$ is $Q$, and $F_{1}$ is the left focus of the hyperbola $C$. Then the minimum value of $\left|P F_{1}\right|+|P Q|$ is $\qquad$ .
5. $2 \sqrt{2}+1$.

Let $F_{2}$ be the right focus of the hyperbola $C$.
Given $\left|P F_{1}\right|-\left|P F_{2}\right|=2 \sqrt{2}$, then
$$
\left|P F_{1}\right|+|P Q|=2 \sqrt{2}+\left|P F_{2}\right|+|P Q| \text {. }
$$

Obviously, when $F_{2}$, $P$, and $Q$ are collinear and point $P$ is between $F_{2}$ and $Q$, $\left|P F_{2}\right|+|P Q|$ is minimized, and the minimum value is the distance from point $F_{2}$ to $l$.
It is easy to know that the equation of $l$ is $y= \pm \frac{1}{\sqrt{2}} x, F_{2}(\sqrt{3}, 0)$. The distance from point $F_{2}$ to $l$ is 1.
Therefore, the minimum value of $\left|P F_{1}\right|+|P Q|$ is $2 \sqrt{2}+1$.