The logo for the Chess Stack Exchange (I pulled the image off of the HTML file) is, to put it plainly, horrible. It is a white rook on a blue background, which is very plain and visually unappealing. We need something much cooler like there is for the Worldbuilding Stack Exchange, to name an example. I personally do not have any ideas, but I am hoping someone here knows how to improve it.
Many small sites are awaiting design (not just Chess), and when the site gets a design, it will get a snazzy icon (like Worldbuilding) among other design features.
For the time being, the site uses the default Stack Exchange colors: a kind of sky blue with white text (main site) and black with white text (meta sites). To my knowledge, there is only one exception for CraftCMS to the default color scheme.
There are restrictions to changing the icons, particularly in regards to copyright. Thus, Stack Exchange have been flexible enough to allow custom temporary icons (like the rook icon here) on some sites, but are unable to go beyond that.
I tried making some icons at Islam which I thought looked okay, but got the response:
I'm sorry to act as a blocker but unless this is part of a unicode character set, we can't use custom art, even if y'all create it. I need to be extremely limiting in what I ask our branding team to do so the guidelines are pretty strict. Until this site gets approved for a full site design or we change the process somehow, you'll need to stick to basic (non-emoji) text.
Catija ♦, January 2020
Realistically, I feel the rook icon is about as good as you'll get until full design. (Alternatives would be changing it to a knight or other chess piece, or an abbreviation "Ch" for "Chess"; these are not really concrete improvements.)