To find the equation of the diameter of the circle that passes through the origin, we can follow these steps:

Step 1: Identify the equation of the circle
The given equation is x^2 + y^2 - 6x - 2y - 8 = 0. To express this equation in standard form, we need to complete the square for both the x and y terms.

Step 2: Complete the square for x terms
Rearrange the equation by grouping the x terms together: x^2 - 6x + y^2 - 2y - 8 = 0.

To complete the square for the x terms, we need to add and subtract the square of half the coefficient of x (which is -3 in this case). So, we have:
x^2 - 6x + (-3)^2 - 9 + y^2 - 2y - 8 = 9 - 9.

Simplifying this equation gives us:
(x - 3)^2 + y^2 - 2y - 17 = 0.

Step 3: Complete the square for y terms
Similar to step 2, we need to complete the square for the y terms. To do this, add and subtract the square of half the coefficient of y (which is -1).
(x - 3)^2 + y^2 - 2y + (-1)^2 - 1 - 17 = 9 - 9 - 1.

Simplifying this equation gives us:
(x - 3)^2 + (y - 1)^2 - 19 = 0.

Step 4: Identify the equation of the circle
From the equation above, we can see that the equation of the circle is:
(x - 3)^2 + (y - 1)^2 = 19.

Step 5: Identify the center of the circle
Comparing the equation of the circle to the standard equation (x - h)^2 + (y - k)^2 = r^2, we can identify the center of the circle as (h, k). In this case, the center is (3, 1).

Step 6: Find the diameter
Since the diameter of a circle passes through its center, we can use the center point (3, 1) and the origin (0, 0) to find the equation of the diameter.

The slope of the line passing through the two points is given by:
m = (y2 - y1) / (x2 - x1) = (1 - 0) / (3 - 0) = 1/3.

Using the point-slope form of a line, we have:
y - y1 = m(x - x1),
y - 0 = (1/3)(x - 0),
y = (1/3)x.

Step 7: Identify the equation of the diameter
The equation of the diameter passing through the origin is y = (1/3)x.

In summary, the equation of the diameter of the circle x^2 + y^2 - 6x - 2y - 8 = 0 that passes through the origin is y = (1/3)x.