A friend sent me this question and the answer is B, I want to know how is it correct to use "she have" with grammatical explanation please.?

B is the best choice because:

"Jane's pale face SUGGESTED…" :(Past Tense verb, so…

"…that she WAS ill;" :Past Tense verb

"and her parents suggested" : With verbs like 'suggest, recommend, insist' and adjectives like 'important, essential, imperative, crucial, vital', we can use the subjunctive, if this is formal; or, less formal, we use 'should + infinitive'.

We use the subjunctive mainly when talking about events that are not certain to happen. For example, we use the subjunctive when talking about events that somebody:
• wants to happen
• anticipates will happen
• imagines happening
•
... to express intention or proposal about the future.

The problem is, would a parent be formal with their daughter?! Maybe if the DOCTOR suggested 'have a blood test'.

This is a poor test item!!!

'Have' could be used and be grammatically correct, but I feel that 'has' would be more appropriate but if preceded by 'should' then it would be 'have'.:-

I have : I should have
You have : You should have
She has : She should have
He has : He should have
We have : We should have
They have : They should have

is the declension. Have changes from has since the addition of 'should' changes it to the future tense from the present tense.