Fabric-ish patterns are close to my heart. French Stucco to the rescue.
Source Christopher Buecheler
This light background pattern has a texture of "frozen" surface with diagonal stripes. Here's an yet another addition to the collection of free website backgrounds.
Source V. Hartikainen
Sometimes simple really is what you need, and this could fit you well.
Source Factorio.us Collective
One more updated pattern. Not really carbon fiber, but it’s the most popular pattern, so I’ll give you an extra choice.
Source Atle Mo
It was called Navy Blue, but I made it dark. You know, the way I like it.
Source Ethan Hamilton
A seamless pattern formed from a square tile. The tile can be retrieved by selecting the rectangle in Inkscape and using shift-alt-i.
Source Firkin
A comeback for you: the popular Escheresque, now in black.
Source Patten
A frame using leaves from an image that was uploaded to Pixabay by mayapujiati
Source Firkin
A free seamless background image with a texture of dark red "canvas". It should look very nice on web sites.
Source V. Hartikainen
This beige background pattern resembles a concrete wall with engravings or something similar to it.
Source V. Hartikainen
It almost looks a bit blurry, but then again, so are fishes.
Source Petr Šulc
White fabric looking texture with some nice random wave features.
Source Hendrik Lammers
Seamless pattern the tile for which can be had by using shift-alt-I on the selected rectangle in Inkscape.
Source Firkin
Used the 6th circle pattern designed by Viscious-Speed to create a print that can be used for card making or scrapbooking. Save as a PDF file for the best printing option.
Source Lovinglf
The image depicts a seamless pattern of Japanese Edo pattern called "kikkou-matsu" or "亀甲松" meaning " tortoiseshell-pinetree".The real pinetree is like this: https://jp.pinterest.com/pin/500744052301065077/
Source Yamachem
Seamless Background For Websites. It has a texture similar to cork-board.
Source V. Hartikainen
Bigger is better, right? So here you have some large carbon fiber.
Source Factorio.us Collective
Seamless pattern the tile for which can be had by using shift-alt-I on the selected rectangle in Inkscape.
Source Firkin
Clover with background for St. Patrick's Day. Add to a card with a doily, ribbon, a leprechaun or other embellishments.
Source BAJ
Bit of a strange name on this one, but still nice. Tiny gray square things.
Source Carlos Valdez
Just to prove my point, here is a slightly modified dark version.
Source Atle Mo
Design drawn in Paint.net, vectorised using Vector Magic and finished in Inkscape.
Source Firkin