"Assign numbers to each patient and then use a random number generator to choose one hundred patients to survey" is the best option.
It is because option 2, 3, and 4 do not have truly randomized surveys, as randomization is important in surveys.
In option 2, the time might affect the patients moods (grumpy in the mornings and/or night) and might give incorrect data.
In option 3, the first 100 patients in alphabetic order do not fully represent all of the patients the doctor has, and might give incorrect data.
In option 4, the nurse might just choose the 100 most cheerful patients and this might give incorrect data.
So, number one, "assign numbers to each patient and then use a random number generator to choose one hundred patients to survey." is the best option.
Hope this helps!