What is a name for a barn loft?

A barn hay loft is also called a hay mow (or haymow; this word is pronounced like 'wow').

In the 1800s, wagons full of hay pulled up on a higher level of ground beside the 2nd story of the barn. Men pitched hay bales through a large door down into the hay mow. Men in the hay mow stacked the bales so more would fit, until the mow was full. Note: Hay bales must be completely dry before being put in the mow. If not, heat can build up inside the bale and spontaneously combust. The fire can burn down the whole barn.

If the farm grew wheat or barley and the barn had a thresher on the first floor, they pitched the grains into the mow. Men on the mow level would then pitch the grain down to the first floor where other men added it to the thresher (or it was pitched directly on top of the thresher where the grains would fall into the thresher, or get a little help from a pitchfork). A single horse, mule, or ox walked in a circle to turn the grindstone in the thresher. During threshing, a cloud would form in the air inside the barn from grain dust-- if it got in the lungs it could cause pulmonary (lung-breathing) problems.

Source: The farm and huge barn my 5th great-grandfather built in the 1840, and descriptions from the woman born 1910 who grew up on the same farm and witnessed these age-old farming processes. She showed me around the hay mows--a huge floor space of at least 30'x30' and over 10-20 feet high. I could still see the circular markings from hooves around where the thresher sat just inside the 2nd level doorway. A trap door on the 2nd level let farmers throw hay down to the ground level where milking cows stayed during the winter.

The name for a barn loft is hay loft.

As far as I am aware, there are no alternative names for the loft of a barn aside from "hay loft".