Hi Michael,
I would have to agree with Christian on this one as "fabric scrap" is reductive in my opinion; there is still so much information that can be gleaned from a textile, even in fragmentary form, that may not be directly related to its function, which may or may not be known to you at this point in time. Thus I would do as Michelle has suggested and choose a more appropriate term and then classify it yourself. That said, I would try to keep any deviations from Nomenclature to the same standard lexicon--I'm a big fan of Getty's Art and Architecture Thesaurus, which I find lends itself better to various textile manufacturing processes and products, among other things.
Terms from Getty I think would be relevant for you include:
Textiles (visual works): General term for carpets, fabrics, costume, or other works made of textile materials, which are natural or synthetic fibers created by weaving, felting, knotting, twining, or otherwise processing. For works of art or high craft that employ textile as a medium, prefer "textile art (visual works)."
Swatches: Small pieces of material such as textile, leather, or paper, either individual pieces or groups of them bound together, especially when issued by a single manufacturer. (I prefer this definition to Nomenclature's as it does not constrain swatches to the realm of merchandising, but does allow for it to be an option)
From Nomenclature I would continue to use Fragment, but just be careful in your distinction between a swatch and a fragment. My sense is that swatches are intentionally small in size and meant only to serve as a sample of a larger material, (particularly when used as merchandising tool/salesman's samples, or collected in bound books for study as I have seen) whereas fragment suggests to me that the object in question is understood to have been part of a larger produced work at one point.
If the object is lace, even in fragmentary form, I would continue to use Lace from Nomenclature, as you are in describing the technique as much as the finished product.
And in regards to lengths of woven cloth (particularly where one or both selvages are present) you could use Yardage from Getty; if the textile is non-woven there is also "Cloth". Nomenclature 4.0 does have Cloth Bolt, but in my mind this should reserved for objects where the textile is actually a bolt or roll of cloth.
Best of luck!
Erica