With the endosymbiotic hypothesis in mind, what structure within modern-day chloroplasts is likely derived from the cytoplasm of an ancestral cyanobacterium that took up residence within a eukaryotic cell?

Inner membrane of the chloroplasts

Explanation:

The present day eukayrotes are all considered to be derived from its original ancestor – the cyanobacteria.

Earlier, photosynthesis by the first photoautotrophs took place utilizing hydrogen sulphide as the electron donor. However, it was later when the cyanobacteria, which were originally residing in the mitochondria of an eukaryotic cell under an endosymbiotic relationship, developed into true chloroplasts, the use of water as electron donor to perform photosynthesis began.  

The cyanobacteria were aerobic in nature and required oxygen to survive. They evolved the chloroplasts covered with an external protective membrane and an internal membrane. It is the internal membrane which contains all the necessary organelles or components necessary for photosynthesis like thyllakoids, stroma etc which helped them to utilize water as an electron donor during photosynthesis like all the eukaryotes.