What to do if my son wants a divorce and his wife refuses to leave?

You have a mix of questions-- so I'm going to break them up in a couple of different parts.

The first issue is getting her out of the house.  Since neither of them own the house, then the ability to force one or both out would depend on the lease agreement.  Lease agreement's usually provide that non-payment is a basis for eviction.  If there was never a lease agreement, but still an agreement of rent in exchange for living there, then grandmother would have the right to file for eviction--- and quite frankly should.  Many lower courts have some simple forms that you may be able to get access to in order to file the eviction without an attorney. 

Another option to get her out is for him to file for divorce and request temporary orders which grant him exclusive use of the residence.  This may or may not be effective because courts don't generally like to disrupt a mother and child for a joint household -- especially for a young child.  Your better option is grandmother filing for eviction because a divorce court can't force a third party to keep a non-paying renter.

Considering there has already been a CDV, he really does need an attorney to help work him through the process so that he doesn't get taken advantage of during the divorce process.  The courts generally do not appoint attorneys in divorce actions.  However, you or your son should do some shopping for attorneys and services before he throws in the towel.  First tactic:  talk to a couple of attorneys who do family law and are willing to accept payment plans or credit cards.  Attorneys have been hit by the ecomony as well, and are much more flexible with their payment options than they used to be.  Considering that your son is working, he may just have to find a lawyer that will accept payments.  Many through the Attorney Pages program offer similar options.   Second tactic:  google or research non-profit legal organizations.  Use terms like "legal aid", "legal services" and "non-profit legal programs" and include your state and county.  Try county wide then state wide so that you can focus on the ones in closest proximity to your son.  Third tactic:  call the district clerk and see if the local bar association offers any local legal clinics that your son can attend for advice, and possibly representation.

As you are looking for an attorney, focus on ones that know family law.  A family law attorney will understand the complexities of your son's case the best.  Try to find one that you are both comfortable talking to.  Divorce means talking about the good, bad, and ugly.... if you're not comfortable with the attorney, then you can't communicate everything they need to know to effectively represent you.

He does not have to wait to get her out of the house before he files. Visit with an attorney regarding the time.... however, he may want to move in with ya'll temporarily until she's evicted to avoid any future allegations of abuse.  If he's not there, it reduces the potential for any other issues while the divorce is ramping up.

Your last question was regarding child support.  If the court orders him to pay, then generally it will be income based.