Inside the circle with center $O$, points $A$ and $B$ are marked so that $OA = OB$. Draw a point $M$ on the circle from which the sum of the distances to points $A$ and $B$ is the smallest among all possible.
1. **Identify the given conditions and the goal:**
   - We have a circle with center \( O \).
   - Points \( A \) and \( B \) are inside the circle such that \( OA = OB \).
   - We need to find a point \( M \) on the circle such that the sum of the distances from \( M \) to \( A \) and \( M \) to \( B \) is minimized.

2. **Use the property of ellipses:**
   - The sum of the distances from any point on an ellipse to its two foci is constant.
   - The point \( M \) on the circle that minimizes the sum of the distances to \( A \) and \( B \) can be found using the optical property of ellipses: the normal to the ellipse at any point bisects the angle between the lines joining the point to the foci.

3. **Apply the optical property:**
   - Since \( OA = OB \), the line \( MO \) (where \( M \) is the point on the circle) will bisect the angle \( \angle AMB \).
   - This implies that \( \angle AMO = \angle BMO \).

4. **Cyclic quadrilateral property:**
   - Given \( OA = OB \), the quadrilateral \( AOBM \) is cyclic (i.e., it can be inscribed in a circle).
   - Therefore, \( M \) lies on the intersection of the circle passing through \( A \), \( O \), and \( B \) with the original circle.

5. **Determine the specific point \( M \):**
   - If the circle passing through \( A \), \( O \), and \( B \) lies inside the original circle, the closest point \( M \) would be where the ray \( ON \) (where \( N \) is the midpoint of \( AB \)) intersects the original circle.

6. **Construct the point \( M \):**
   - Draw the circle passing through \( A \), \( O \), and \( B \).
   - Identify the intersection points of this circle with the original circle.
   - The point \( M \) that minimizes the sum of the distances \( MA + MB \) is the point where the ray \( ON \) intersects the original circle.

The final answer is the point \( \boxed{ M } \) on the original circle where the ray \( ON \) (with \( N \) being the midpoint of \( AB \)) intersects the circle.