Step 1: Use the reflection property.
A mirror line $L$ is the perpendicular bisector of the segment joining any point to its image. So $L$ perpendicularly bisects $PQ$ with $P(2,3)$ and $Q(4,5)$.
Step 2: Find the midpoint of $PQ$.
\[ M=\left(\frac{2+4}{2},\frac{3+5}{2}\right)=(3,4). \]
Step 3: Find the slope of $L$.
Slope of $PQ$ is $\dfrac{5-3}{4-2}=1$, so the mirror line has slope $-1$.
Step 4: Write the equation of $L$.
Through $(3,4)$ with slope $-1$: $y-4=-(x-3)$, i.e. $x+y-7=0$.
Step 5: Reflect the origin in $L$.
For line $x+y-7=0$ the reflection of $(0,0)$ uses $x'=x-\dfrac{2\cdot1\cdot(0+0-7)}{1^2+1^2}=0+7=7$, and similarly $y'=7$.
Step 6: State the image.
Thus $R(0,0)$ maps to $(7,7)$, which is option 4.
\[ \boxed{(7,7)} \]