Is there a name for the truncated bedspread that some hotels put on the bottom 25% or so of the bed?

Solution 1:

It's called a bed scarf or sometimes partial coverlet or possibly (as StoneyB notes in comments) a bed runner. From The Linen Gallery Blog:

A bed scarf is just that — a two foot by 8 foot piece of fabric or a knit throw that is meant to go at the foot of the bed. It is suppose to be functional in that it protects the bedding when someone lays down on top of it with shoes or puts a piece of luggage on it. ("What's a Bed Scarf?", July 1, 2010)

As suggested by this definition, there is an ostensibly practical purpose for the bed scarf; another travel blogger notes that

The first time I noticed a bed scarf was in a cabin on a ship. Since space is tight in a stateroom, most passengers heave their suitcase on top of the bed to unpack. Functional rather than pretty, that ship's bed scarf was a strip of vinyl placed to protect the bedspread from being soiled by dirt that the wheelse [sic] and bottom of luggage may have picked up during transit. (Susan Breslow Sardone, "What is a Bed Scarf? Do You Know Its Purpose?", About Travel, October 25, 2016)

However, they are now very often primarily decorative. The Home Shopping Network's bedding glossary emphasizes the decorative purpose:

Bed Scarf– Fabric accessory that drapes across the bottom of a comforter or other bed covering for a decorative effect

You can see many examples, especially of the decorative variety, with a Google Image Search: