Question with "什么颜色" or "什么颜色的"
...什么颜色 or ...什么颜色的 may both be used. However, their constructed sentence have different analytical structures in the grammatical sense.

In your other question, you may have already seen an example of Chinese grammar, that some words are implied.

The two complete sentences are as follows:

你的眼睛是什麼顏色？ What color are your eyes? 什麼顏色 acts as a noun

你的眼睛是什麼顏色的(眼睛)？ What color(ed eyes) are your eyes? 什麼顏色 acts as an adjective



Here I analyze them using grammatical concepts.

Step 1:

/-Subject-\/-Predicate-\         &lt;--- Step 1
  你的眼睛    是什麼顏色？

/-Subject-\/--Predicate---\      &lt;--- Step 1
  你的眼睛  是什麼顏色的(眼睛)？


Step 2 &amp; 3:

/--Subject--\/-Predicate-\            &lt;--- Step 1
/-M-\   /-H-\/-P-\ /--O--\            &lt;--- Step 2
  你  的 眼睛   是   什麼顏色？

/--Subject--\/-----Predicate------\   &lt;--- Step 1
/-M-\   /-H-\/-P-\ /------O-------\   &lt;--- Step 2
|   |   |   ||   | /--M--\    /-H-\   &lt;--- Step 3
  你  的 眼睛   是   什麼顏色 的 (眼睛)？

M: Modifier
H: Head
P: Predicator
O: Object


Here we see a pattern: (Modifier)的(Head). Modifier is a fragment that describes the Head. For example: 


(你)的(眼睛). Eyes -> whose eyes? -> your eyes
(什麼顏色)的(眼睛). Eyes -> how are the eyes? -> What colored eyes


As you can see, the two questions are asking for different answers. 


你的眼睛是什麼顏色？asks for the color
你的眼睛是什麼顏色的(眼睛)？ asks for the eyes in what color


And so, the conversation should be as follows:

Q: 你的眼睛是什麼顏色？ What color are your eyes?
A: 黑色。 Black.


Q: 你的眼睛是什麼顏色的(眼睛)？ What color(ed eyes) are your eyes?
A: 黑色的(眼睛)。 Black color(ed eyes). 的 is necessary




Side note, you may also reverse the order of the question:

Q: 你的眼睛是什麼顏色? -&gt; 你(的)眼睛的顏色是什麼? (We take out all but the last 的 in common situations, but the modifier behavior is still here.)
A: 黑色。


These two sentences are pretty much the same. Analytically, both are still asking for the color quality of "your eyes".