Because English is a context-dependent language, this sort of artificial question set by unthinking examiners is usually pointless.
My guess is that the intended answer is "who".
However, as has been discussed in the comments, all options are possible but they have different meanings. Here are some examples: If I spent longer I could make them more convincing.
I want to be liked and loved for who I am inside. ---> I want to be liked and loved for the person I am inside.
I want to be liked and loved for where I am inside. ---> I want to be liked and loved for the stage I have reached in my personal development.
I want to be liked and loved for what I am inside. ---> I want to be liked and loved for the bundle of emotions that I am inside.
I want to be liked and loved for how I am inside. ---> I want to be liked and loved for my interior essence.