To find the measure of side \( RO \), we need to understand the relationship between the original rectangle \( OPQR \) and the dilated rectangle \( O'P'Q'R' \). The dilation is done using a scale factor of \( 6 \).
When a shape is dilated, each side of the shape is multiplied by the scale factor. Therefore, if we denote the length of side \( RO \) as \( x \), after dilation, the corresponding side \( R'O' \) will be given by:
\[ R'O' = \text{scale factor} \times RO \]
Substituting the values we know:
\[ 96 = 6 \times x \]
To find the value of \( x \), we solve for \( x \):
\[ x = \frac{96}{6} = 16 \]
Thus, the measure of side \( RO \) is:
\[ \boxed{16} \]