**Core Concept**
The question is testing the ability to identify a specific bacterial pathogen based on its morphological characteristics, growth requirements, and clinical presentation. The presence of Gram-negative coccobacilli that grow on chocolate agar but not on blood agar, along with the clinical presentation of fever, vomiting, neck rigidity, and CSF findings, is a key to identifying the causative agent.

**Why the Correct Answer is Right**
The Gram stain shows the presence of Gram-negative coccobacilli, which is a characteristic feature of *Haemophilus influenzae*. The growth of bacteria only on chocolate agar and not on blood agar is also a diagnostic clue, as *H. influenzae* requires the X and V factors present in chocolate agar for growth. The CSF findings of polymorphs more than 2000/ml, protein 100 mg/dl, and glucose 10 mg/dl are consistent with bacterial meningitis, which is a common presentation of *H. influenzae* type b (Hib) infection.

**Why Each Wrong Option is Incorrect**
* **Option A:** This option is incorrect because *Neisseria meningitidis* is a Gram-negative diplococcus, not a coccobacillus.
* **Option B:** This option is incorrect because *Streptococcus pneumoniae* is a Gram-positive coccus that grows on blood agar, not chocolate agar.
* **Option C:** This option is incorrect because *Escherichia coli* is a Gram-negative rod, not a coccobacillus, and it grows on both chocolate agar and blood agar.

**Clinical Pearl / High-Yield Fact**
It's essential to remember that *Haemophilus influenzae* type b (Hib) is a common cause of bacterial meningitis in children, and the introduction of Hib conjugate vaccine has significantly reduced its incidence.

**Correct Answer:** D.