3. Let the three sides of a right-angled triangle be $a$, $b$, and $c$ ($c$ being the hypotenuse), with the perimeter being 6. Then the minimum value of $c$ is

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3. $6(\sqrt{2}-1)$.

Let the side $a$ of a right-angled triangle be opposite the angle $\alpha$. Then $0<\alpha<\frac{\pi}{2}$.
Thus, $a=c \sin \alpha, b=c \cos \alpha$.
Also, $a+b+c=6$
$\Rightarrow c(\sin \alpha+\cos \alpha+1)=6$
$\Rightarrow c\left[\sqrt{2} \sin \left(\alpha+\frac{\pi}{4}\right)+1\right]=6$
$\Rightarrow c=\frac{6}{\sqrt{2} \sin \left(\alpha+\frac{\pi}{4}\right)+1}$.
Since $\sin \left(\alpha+\frac{\pi}{4}\right)$ reaches its maximum value when $\alpha=\frac{\pi}{4}$
Therefore, when $\alpha=\frac{\pi}{4}$, $c$ reaches its minimum value $6(\sqrt{2}-1)$.