why were these cells named as such?
Because of the bacteria prokaryotes and organisms with nuclei.
Eukaryote
 Chatton had proposed the two terms, calling the bacteria prokaryotes and organisms with nuclei in their cells eukaryotes. 
does eukaryotes have specialized structures or internal compartments Yes, they have long slender motile cytoplasmic projections, called flagella, or similar structures called cilia. how are they related to the membrane system? The endomembrane system is composed of the different membranes that are suspended in the cytoplasm within a eukaryotic cell.
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Eukaryote [SEP] History of the concept
The concept of the eukaryote has been attributed to the French biologist Edouard Chatton (1883–1947). The terms prokaryote and eukaryote were more definitively reintroduced by the Canadian microbiologist Roger Stanier and the Dutch-American microbiologist C. B. van Niel in 1962. In his 1937 work "Titres et Travaux Scientifiques", Chatton had proposed the two terms, calling the bacteria prokaryotes and organisms with nuclei in their cells eukaryotes. However he mentioned this in only one paragraph, and the idea was effectively ignored until Chatton's statement was rediscovered by Stanier and van Niel. In 1905 and 1910, the Russian biologist Konstantin Mereschkowski (1855–1921) argued that plastids were reduced cyanobacteria in a symbiosis with a non-photosynthetic (heterotrophic) host that was itself formed by symbiosis between an amoeba-like host and a bacterium-like cell that formed the nucleus.