To find the length of the other side of the right triangle, we can use the Pythagorean theorem, which states:
\[ a^2 + b^2 = c^2 \]
where \( c \) is the hypotenuse and \( a \) and \( b \) are the lengths of the other two sides. In this case, we know the hypotenuse \( c = 17 \) cm and one side \( a = 15 \) cm. We need to find the length of the other side \( b \).
Plugging the known values into the Pythagorean theorem:
\[ 15^2 + b^2 = 17^2 \]
Calculating \( 15^2 \) and \( 17^2 \):
\[ 225 + b^2 = 289 \]
Next, we can subtract 225 from both sides to solve for \( b^2 \):
\[ b^2 = 289 - 225 \]
\[ b^2 = 64 \]
Now, taking the square root of both sides to find \( b \):
\[ b = \sqrt{64} \]
\[ b = 8 \]
Therefore, the length of the other side of the triangle is \( \boxed{8} \) centimeters.