Conductive materials are potential candidates for developing bone tissue engineering scaffolds as they are nontoxic and can enhance bone tissue regeneration. Their bioactivity can be enhanced by depositing biomineralization in simulated body fluid (SBF). In the current study, a composite electrospun membrane made up of poly(lactic) acid, poly(3-hydroxybutyrate-co-3-hydroxyvalerate), and hydroxyapatite was fabricated using an electrospinning method. The fabricated membranes were dip-coated with a conductive polymer solution, poly(3,4-ethylenedioxythiophene) poly(4-styrenesulfonate), to induce conductivity. Characterization of the membranes based on characteristics such as morphology, chemical bonding, and wettability was conducted using scanning electron microscopy, field emission scanning electron microscopy, energy-dispersive X-ray spectroscopy, attenuated total reflectance Fourier transform infrared spectroscopy, and contact angle measurement. From the results, biomineralization of both coated and noncoated composite membranes was observed on the surface of nanofibers after 21 days in SBF. The membranes provide a superhydrophilic surface as shown by the contact angle. In conclusion, this biomimetic electrospun composite membrane could be used to further support cell growth for bone tissue engineering application.