“Say to” or “tell”

This is a discussion which I started on the site italki.com about the English Grammar:

There is a question in the test:
Do you know who she is? No, she didn´t ___ her name.

According to Murphy ‘English Grammar in Use’:
‘If you say who somebody is talking to, use tell:
- Sonia told me that you were in hospital. (not Sonia said me)
But you can 'say something to somebody':
- What did you say to the police?’

Does it mean that there are two right answers: "say to me" and "tell me"?

According to the author of the test - the right answer "tell me". Why?

Comment 1:

Right answers could be "say" or "tell me/us". If the exercise requires two words to fill the gap, then "tell me/us" is the only option. If the ex. does not specify this, then it is an unfair one.

It is correct that you use "say + to + person" and "tell + person".

But you can extend these structures like this:
"say + something (e.g. her name) to + person"
"tell + person + something (e.g. her name)".

You can omit the last idea in the structures where it is obvious from the context:
"say + something e.g her name"
"tell + a person e.g. me"

Comment 2:

You "tell" someone something.
You "say" something to someone.

I want to tell you what I heard last night.
Jack says the party starts at 8pm.

Comment 3:

Further to previous explanation, I would like to add that, in this case, the answer could be 'give' : 'she didn´t give her name'.

This is a 'one-off' specific case which only applies in this context e.g. 'give me your name' i.e. 'tell me your name', and is only used verbally.

Comment 4:

it is not a very good test. IT IS UNFAIR

you can insert either"say" or "tell" her name into the missing gap.But say would be the most natural.

to make it correct for tell it would be.

Do you know who she is? No, she didn't tell us/me her name.

tell is used to say she passed on the information by saying or other means

Say is used to say she passed on information by saying speaking.