Seamless pattern made from a tile that can be obtained in Inkscape by selecting the rectangle and using shift+alt+i.
Source Firkin
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
One more from Badhon, sharp horizontal lines making an embossed paper feeling.
Source Badhon Ebrahim
Zerro CC tillable texture of stones photographed and made by me. CC0
Source Sojan Janso
A repeating background with dark brown stone-like texture and abstract pattern that looks like tree trunks.
Source V. Hartikainen
Prismatic Curved Diamond Pattern 8 No Background
Source GDJ
"Beige Stone", Tileable Texture.
Source V. Hartikainen
And some more testing, this time with Seamless Studio. It’s Robots FFS!
Source Seamless Studio
Bright gray tones with a hint of some metal surface.
Source Hendrik Lammers
The texture of this background image has some similarities with leather, and it's colored in a dark brown color. So, if you are looking for a dark brown background image for your website, this may be an option for you.
Source V. Hartikainen
A pattern derived from repeating unit cells each derived from part of a fractal rendering in paint.net.
Source Firkin
Gold Triangular Seamless Pattern No Background
Source GDJ
Snap! It’s a pattern, and it’s not grayscale! Of course you can always change the color in Photoshop.
Source Atle Mo
A beautiful dark wood pattern, superbly tiled.
Source Omar Alvarado
A seamless pattern the unit cell for which can be had by selecting the rectangle in Inkscape and using shift+alt+i.
Source Firkin
Seamless pattern the tile for which can be had by using shift-alt-I on the selected rectangle in Inkscape.
Source Firkin
Tiny little fibers making a soft and sweet look.
Source Badhon Ebrahim
A bit simplified version. Although it could be edited out to be simpler. Anyway, this time the tiling is converted to a pattern fill -which is using clipping for the tile's edges.
Source Lazur URH
Prismatic Rounded Squares Grid 4 No Background
Source GDJ
Light honeycomb pattern made up of the classic hexagon shape.
Source Federica Pelzel
Seamless pattern formed from a tile that can be had by selecting the rectangle in Inkscape and using shift-alt-i.
Source Firkin
From a drawing in 'Hundert Jahre in Wort und Bild', S. Stefan, 1899.
Source Firkin
Sweet and subtle white plaster with hints of noise and grunge.
Source Phil Maurer