What hotels that are relatevely cheap and near Edinburgh Train Station?.

None of them?

You're talking about one of the world's major tourist destinations at
the height of its season (the Edinburgh Festival is about to start).
Most accommodation is already booked up.

Try the tourist information office at Waverley Station (part of
"visitscotland" on the Web) - they are utter crap at anything you
might expect of a tourist board except for booking accommodation,
but that they can do (except that they won't tell you about the
places that don't fit their weird grading scheme and pay them to
be listed).

You could try a backpacker hostel (there are too many to list, some
near each of the two railway stations) but it'll probably take a
terrorist bombing or an outbreak of Ebola to create you a vacancy.

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