What do you want to eat?
"I don't know, I don't care, anything is fine." --> NOT COURTEOUS.
When do you want to schedule that lunch meeting? I can do Wednesday, Friday, or the following Monday.
"I don't care, any day is good." --> NOT COURTEOUS.
They seem courteous, but you're really offloading the decisionmaking back to the other person. And the thing is, maybe you don't do this, but a lot of people will say they're totally flexible but then dislike the first option suggested and only then do their thinking and propose something.
Much better:
What do you want to eat?
"I really don't have a preference, so maybe Italian at Belucci's? But really, whatever you want." --> COURTEOUS.
When do you want to schedule that lunch meeting? I can do Wednesday, Friday, or the following Monday.
"Any of those work. How about Friday?" --> COURTEOUS.
Seriously, try it out. If you really don't care, arbitrarily pick one. Go ahead and let the other person know you're flexible, but offloading decision making back on someone by being totally flexible isn't actually courteous.