Hcfyk

Yes, rather her do is grammatical.

Compare rather me do.

A pronoun in the objective case is licensed by the verb rather.

This is also valid:

What would you rather she say?

Here are two (of many more) ways to express the sentiment:

1. What would you rather have her do?

and

2. What would you rather she did?

It is difficult for one who is not a trained linguistics/grammar expert to sort this one out.
However my native ear tells me that all of the following are idiomatic, and in daily use in GB.

What would you rather she do. (Some might argue that does could be substituted here, but I certainly prefer do)

What would you rather her do.

What would you rather have her do.

Turning to old reliable Will Shakespeare (in As You Like It)

 ...Truly, I would 
The gods had made thee poetical.

The verb in this problem is "would". "Rather" is simply an adverb, with no particular consequence.

An Elizabethan-ish dialogue:

I would she had done something else.
What would you she had done?
But soft! She is still doing that terrible thing. 
What would you she were doing?
Ah, she has stopped and asks now for guidance.
What would you she do?

Put in "rather," well, as you like it.

"What would you rather she do?" is correct, because it requires present tense, singular.

"Does" is for ongoing action, so it would not be correct. "Do" is for singular subjects and single actions, with auxiliary verbs (can, should, etc). "Can she do it?" not "can she does it?" ("She does it all the time," however.)

"What did she do?" and "What should she do?" are correct in subject-verb agreement and tense.

Using "her", the objective form of "she," is not correct either.

She - does something (action). "She goes to bed early." "She can fly high."

(Action) -- to her (object of verb). "He read it to her."

Examples of subject + would rather

• I'd rather he leave now. Or: I'd rather he left now.


• I'd rather he do the work now. Or: I'd rather he did the work now.



• What would you rather she do? Or: What would you rather she did?




• I'd rather he left now. [also]



• I'd rather he did the work now. [also]


• What would you rather she did? [also]

she does and her do don't work here.

The third-person singular s or es morpheme is not used after "would rather": subject ][I,we, etc.] would rather.

The choices are present simple without the third person singular morphemes (s or es) or simple past.

Real World Examples:

Arsenal can’t win for losing
20
What would you rather they do: something or nothing?What would you rather they do?

"Seriously - read what you typed. By far the biggest element of fuel is tax. The government should reduce the tax. And pay for what how???? Sadly the only means of raising revenue is.... tax. What would you rather they do? Shut your local hospital to pay for your drive down the motorway? Jeez."

what would you rather they do?

"The public has indicated that they would rather he do this than increase taxes."
would rather he do

Followed by simple past:

But Labour Leader Jeremy Corbyn told ITV News that he would "have a discussion" with Rosena Allin-Khan, who is a shadow minister, about her appearance at the rally: "She's entitled to her point of view. I would rather she and every other Labour MP spent today and tomorrow and Tuesday concentrating solely on making sure we defeat this deal."

would rather plus simple past

Firstly, according to 3,000 voters asked by the website Politico, 47 per cent of voters want Theresa May to compromise and reach a deal with the European Union, compared to 35 per cent who would rather she walked away without one. Indeed, 53 per cent of voters to 47 would rather we stayed in the European Union than exited with no deal in place.
https://www.independent.co.uk/voices/theresa-may-conservatives-brexit-no-deal-politics-westminster-leadership-a8618456.html

This is called causative.

The subject of the action/sentence is you, who is the cause of an action of a second person. Even though it might seem that the verb is an action that she will do, this is not the case. The action is done by you. The action is to make her do or have her do. In your case, it has been shortened to her do. This is the verb that you do.

I am not sure if the omission of the verb make or have is considered undesirable. It is definitely used as you wrote it.

It is very common to express causative value using make/have + noun + bare infinitive. The bare infinitive is the action that noun will do. In your case the do is the bare infinitive.

But since the sentence is really expressing an action of its subject you, it would be the verb make/have the one that carries the declination by person. For example:

She makes her do.

Because some are confused about it. - There are many ways in which a sentence can have causative value. Your example is but one of them. Causative value can be constructed also using have + noun + past participle as in

She had him killed.