Can a family member withdraw an assault charge after his brother's case was picked up by the Harris County Texas Grand Jury?

No. Once a charge is accepted by the DA's office they are the only agency (other than the judge in certain situations) that can dismiss the prosecution. What you can do is speak to your brother's attorney. Depending upon what you have to say, you might be able to help your brother's lawyer get the case dismissed. You may also consider going directly to the DA and speaking to whomever is handling the case. I think you going to speak to the DA is the worst option. If you don't know exactly what helps and how to use what you have to say in a helpful way, you may end up doing more harm than good. In fact, that's usually what happens. Find him a good lawyer so that, if it can be done right, it is done right the first time.