Concept:
Recurrent infections with Neisseria species are classically associated with deficiency of the terminal complement pathway, especially C5 to C9. These components form the membrane attack complex.
Step 1: Identify the organism.
The patient has recurrent Neisseria gonorrhoeae infection. Recurrent infections by Neisseria suggest a defect in complement-mediated bacterial killing.
Step 2: Recall the role of terminal complement.
The terminal complement components C5, C6, C7, C8, and C9 form:
\[
\text{Membrane Attack Complex (MAC)}
\]
MAC is important for killing Gram-negative bacteria, especially Neisseria.
Step 3: Select the best investigation.
To detect a defect in this pathway, the best test is:
\[
\text{Terminal complement } (C5-C9) \text{ assay}
\]
Therefore, the correct answer is Terminal complement (C5--C9) assay.