I lived on Warrender Park Terrace when I was a student, and there were similar flats nearby in Melville Place as well as Melville Terrace. They were certainly built before the 1920s, so would have been there in 1930. And no, there are no houses on Melville Drive. There are also Melville Street, Melville Street Lane and Melville Crescent, which are several decades older than Melville Place and Terrace, and in a different part of the city.
You have set me wondering, now, where students actually did live in the 1920s. When I was a student it was normal for a group of students to share occupancy of a flat, but I am not so sure that that was the norm in the 1920s. I suspect that most students from out of town would have lived in lodgings or 'digs', that is, rented a room in a house or flat where the landlady also supplied meals. I actually lived in digs for most of my university years, only moving into a shared flat in my final year.
Also, when asked for my address for any purpose other than university-related things I always used my parents' home address, because I regarded the digs and flat addresses ase temporary.
Do you know where your cousin's parents lived at the time? If in Edinburgh, she may have lived with them at home, in which case you should be able to find them in the valuation rolls and registers of electors.
If she was in a lodging house, you might find the name of the landlady and of any lodgers eligible to vote. A street directory might also list the name of the landlady if it was a lodging house rather than just someone letting one or two bedrooms to students.
The valuation rolls and registers of electors are available in the National Archives of Scotland, the National Library of Scotland and the Edinburgh Centrel Library. The NAS has digitised a few of them, but most have to be consulted in the form of large tomes.
