**Core Concept**
The clinical presentation of high-grade fever, headache, disorientation, projectile vomiting, neck rigidity, and positive Kernig's sign is suggestive of meningitis, an infection of the protective membranes covering the brain and spinal cord. Gram staining of cerebrospinal fluid (CSF) showing gram-negative cocci indicates a specific type of bacterial meningitis.

**Why the Correct Answer is Right**
The presence of gram-negative cocci in the CSF is indicative of Neisseria meningitidis, a common cause of bacterial meningitis, especially in children and young adults. N. meningitidis is a highly infectious and virulent bacterium that can cause severe meningitis and sepsis. The bacterium's ability to adhere to and invade the epithelial cells of the nasopharynx and invade the bloodstream makes it a significant pathogen.

**Why Each Wrong Option is Incorrect**
**Option A:** This option is incorrect because Streptococcus pneumoniae, a gram-positive diplococcus, is another common cause of bacterial meningitis, but the presence of gram-negative cocci in the CSF makes this option less likely.

**Option B:** This option is incorrect because Haemophilus influenzae type b (Hib) is a gram-negative bacillus, not a gram-negative coccus, and is a common cause of meningitis in children, but the gram staining results do not match.

**Option C:** This option is incorrect because Escherichia coli (E. coli) is a gram-negative bacillus, not a gram-negative coccus, and is a common cause of urinary tract infections and neonatal meningitis, but the gram staining results do not match.

**Clinical Pearl / High-Yield Fact**
Neisseria meningitidis can cause outbreaks of meningitis, especially in crowded settings such as schools and military barracks, due to its high infectivity and ability to spread through respiratory droplets.

**Correct Answer:** D.