We need a steady rest library, with pics and dimensions.
Identifying a mystery steady rest or finding a missing steady rest is a daunting and usually unfruitful task.
You know, this community could make a go of it. We collectively have the data to make a searchable library.
While there are lots of lathes around, there are far fewer manufacturers, and they tend to have a few families that have similar dimensions.
There are three easily-measured critical dimensions that will serve to immediately eliminate the unlikely, yielding a reasonable number of candidates. The three are the spacing between bedway pad centers, the distance between the flat bedway plane to the spindle rotation centerline, and the Vee included angle (180 degrees if flat). If one bedway pad has a Vee groove, use the center of the Vee.
Many Vee grooves have a 90 degree included angle, and this can be verified with a combination square or the like. Clausing uses 70 degrees. I think Colchester also uses 70 degrees, or did. I've heard that Harrison used 70 or 75 degrees. And so on.
It's easy to measure the included angle of a Vee groove, using some drill blanks and a dial indicator - one measures how deep in the groove the two drill blanks rest as a function drill blank diameter, yielding the rise over run ratio, which will be the tangent of the included angle.
Math: Treat each measurement point {lathe make and model; width, height, angle} as a point in 3D space. When an unknown steady appears, take its measurements and compute the distance from the new guy to all the old guys currently in the database. Sort by distance, report the closest five or ten candidates, and just how close they are. With luck there will be a big gap between number one and number two, and number one will be the clear winner.