I asked a Thai friend and she said both these sentences have the same meaning:

ต้องการสั่งอะไรดีคะ?

ต้องการสั่งอะไรคะ?

Does that mean that ดี has no meaning sometimes? I know that usually it means good or fine. But in this sentence it seems to have no purpose.

While I might agree that both of your sentences essentially mean the same thing, I wouldn't agree there isn't any difference at all in the meaning. Adding "dee" in my view makes the sentence a bit more polite.

If a western waiter said "what would you like?" or "what good things would you like to order?", most would argue that essentially the waiter is asking the same thing; however, to me, the second one is slightly more pleasant or more polite.

But, yea, not much difference.

ดี in this case is used to ask someones opinion on what to do

In English it's usually translated as would or should.

In your example, since ต้องการ can also be translated as "would like", there isn't much of a difference in translation.

Still, if you leave ต้องการ out, the 2 sentences become really different:

สั่งอะไร - what did you order?/ what are you ordering?

สั่งอะไรดี - what should I/we order?

In the link from t-l.com you posted there are a couple of example sentences that use ดี in this way:

ทานที่นี่หรือที่บ้านดี - "Would you like to eat here or would you like this to go [take away]?

เวลาไหนจะดีที่สำหรับโทรหาคุณคืนนี้ - What would be the best time to call you tonight?

แต่ยังไม่รู้ว่าจะไปไหนดี - But, I still don't know yet where I [we] should go.

ฉันจะทำอะไรก่อนดี - What should I do first?

คืนนี้ไปเที่ยวไหนกันดี พวกเรา - Where should we hang out tonight?

ต้องการสั่งอะไรดีคะ?

The use of the word "ดี" in the above sentence primarily conveys a sense of goodwill/intent of the speaker to meet the listener's needs. Essentially more personable, friendly, welcoming, polite.

It's the same as an English speaking waitperson asking "What can I get for you that's good." versus "What can I get for you."

In the sentence ฉันจะตอบอย่องไงดี? (What's a good way (the best way) to answer?) the use of ดี "dii" conveys the sense that the speaker is looking for the best way to answer the question, again expressing that the speaker has the intent to answer as best they can. Without the use of dii, the meaning is more along the lines of "How should I answer this?"

Anyway, that's my understanding, but interested in any other thoughts others might wish to share.

ต้องการสั่งอะไรดีคะ?

The use of the word "ดี" in the above sentence primarily conveys a sense of goodwill/intent of the speaker to meet the listener's needs. Essentially more personable, friendly, welcoming, polite.

Agree completely. And (just my view) that same thing could be said about all the other listed questions. Each of the listed questions in my view asks for the listener's opinion or response and adding "dee" just makes it more polite/positive.

Can ดี be used in questions where I want someone's opinion but the person who benefits from the answer is myself. I.e. their is no goodwill directed at the listener.

Examples:
1. Let's go to the movies. What should we watch?

2. I'm going to the movies by myself. What do you think I should watch?

In the first example, I am asking the listener for advice and I am doing it for his benefit. I will try and pick a movie that he wants to watch.

In the second example, I am asking someone who may have watched lots movies in the past and knows which ones I might enjoy. In this case, the asker benefits from the response given.

In the second example, would it still be appropriate to use ดี in the question or not?

Edited March 14, 201511 yr by jauggy

Not sure but hopefully somebody with some real expertise might be able to answer your question. Your first example (presuming you're directing it to the listener's benefit) really ought to read (second clause) "what would you like to watch?" Seems to me, though, that "dee" could be added in either your first example or even my version. I'm not sure about the second phrase, sounds a little awkward to my ears.

Can ดี be used in questions where I want someone's opinion but the person who benefits from the answer is myself. I.e. their is no goodwill directed at the listener.

Examples:

1. Let's go to the movies. What should we watch?

2. I'm going to the movies by myself. What do you think I should watch?

In the first example, I am asking the listener for advice and I am doing it for his benefit. I will try and pick a movie that he wants to watch.

In the second example, I am asking someone who may have watched lots movies in the past and knows which ones I might enjoy. In this case, the asker benefits from the response given.

In the second example, would it still be appropriate to use ดี in the question or not?

jauggy:

In example 2, if your intention is to ask, "Do you have any recommendations about movies I might enjoy?" it would be perfectly acceptable to ask

คุณมีคำแนะนำดีๆ ไหม? Do you have any good recommendations (for me)?

Yes that's right. The second example is supposed to be about asking someone for advice that would benefit the asker. Other examples might be

"What bus should I catch to go home?"

"How many sit ups should I do per day to get a six pack?"

"I think Sandy likes me. Should I ask her out?"

Examples of questions that benefit the listener/responder would be:
"What would you like to order?"
"How should I contact you?"
"What time is convenient for you to meet up?"

I found a youtube video that said "dii" is actually added at the end of questions when asking for advice.

It seems like it doesn't matter whether it benefits the asker or the responder. It is likely used in both situations.

I thought that sticking on ดี meant that some information was being asked for,not necessarily good or bad.