I am wondering what is the subtle or delicate difference in meaning between the following?
A. I had my hair blonde
B. I got my hair blonde
In this particular case, where the verb "dyed" is implied (I got my hair dyed blond, I had my hair dyed blond), there is no difference.
More generally, to "get" something usually suggests taking part in some action, while to "have" simply establishes that you possess or control it. In these cases, get and have are used as verbs. Sometimes "have" is used to establish tense (I have owned my car for 10 years), and in this case it does not have (see the usage?) the same distinction.
The word "had" in your first sentence could mean that sometime in the past, your hair was blond, but your hair is no longer blond now.
The word "got" in your second sentence implies that you recently changed your hair color to blond and that your hair is currently blond.
Another possible way to write your sentences would be:
A. My hair was dyed blond.
B. My hair is dyed blond.
The two sentences you said are, as WhatRoughBeast said, ommitting the dyed. If you say "I got my hair [dyed] blond" you are saying that, in the past, you went and had someone dye your hair while if you say "I had my hair [dyed] blond" you are saying that, at some point in the past, your hair was made to be the color blond. They are very similar. This is an example of perfect past, where the verb have in its past form is used with another verb (dyed in this case), to form a statement that says something happened in the past.