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Draper is one of the few stations in Adelaide not named after a current suburb. Draper was a suburb or locality to the north of Largs Bay Estate and south of Taperoo and Osborne, in what is now Largs North. One, as yet, unsourced report suggests Draper was named after a drapery that used to exist nearby (corner of Strathfield Terrace and Victoria Road). In the mid 1980s, the drapery closed and the site has housed a number of different shops since including a pet shop and a fish shop. The shop was recently demolished, to make way for housing.
On 22 August 2024, a 53-year-old man was struck and killed by a northbound train near the station after assaulting staff at a shop and vet on Strathfield Terrace and stealing a packet of sugar. The line was shut down for hours with trains terminating at Glanville and Alberton.