**Core Concept**
H. influenzae meningitis is a serious infection that can lead to significant morbidity and mortality, especially in children. To prevent relapse or complications, it's crucial to investigate for any residual infection or damage before discharging a patient.

**Why the Correct Answer is Right**
The correct investigation is a **lumbar puncture (LP) or cerebrospinal fluid (CSF) analysis**. This is necessary to check for any signs of residual infection, such as elevated white blood cell count, bacterial culture positivity, or abnormal protein levels. A CSF analysis helps determine if the child has achieved complete resolution of the infection, thereby reducing the risk of relapse.

**Why Each Wrong Option is Incorrect**
* **Option A:** Audiometry is not a primary concern in this scenario, as it's more relevant to assessing hearing loss in patients with otitis media or other conditions.
* **Option B:** Serum electrophoresis is not directly related to the management of H. influenzae meningitis.
* **Option C:** An MRI is a valuable diagnostic tool, but it's not the primary investigation required before discharging a child with H. influenzae meningitis.

**Clinical Pearl / High-Yield Fact**
It's essential to remember that a CSF analysis is crucial in assessing the resolution of bacterial meningitis, as it helps guide the decision to discharge the patient safely.

**Correct Answer:** C. MRI.