10. $(1,2]$.

Connect $O C$, intersecting $A B$ at point $D$. Let $\overrightarrow{O D}=m \overrightarrow{O C}$. Then, by the problem, $\overrightarrow{O D}=m \lambda \overrightarrow{O A}+m \mu \overrightarrow{O B}$, and $m \lambda+m \mu=1 \Rightarrow \lambda+\mu=\frac{1}{m}$.

Assume the radius of the circle is 1, and draw $O E \perp A B$, with the foot of the perpendicular at $E$.
Given $\angle A O B=120^{\circ}$, we know $O E=\frac{1}{2}$.
Since $O D \geqslant O E=\frac{1}{2}$, and point $C$ is inside the minor arc $\overparen{A B}$ (point $C$ does not coincide with $A$ or $B$), we have $\frac{1}{2} \leqslant m<1$.
Therefore, $1<\lambda+\mu \leqslant 2$.
10. Let $A$, $B$, and $C$ be three distinct points on $\odot O$, and $\angle A O B=120^{\circ}$, point $C$ lies on the minor arc $\overparen{A B}$ (point $C$ does not coincide with $A$ or $B$). If $\overrightarrow{O C}=\lambda \overrightarrow{O A}+\mu \overrightarrow{O B}(\lambda, \mu \in \mathbf{R})$, then the range of $\lambda+\mu$ is