I would love to see a logo for our site so we have a more distinct and recognizable appearance. In this recent Meta post it says that
If your site's users feel like there are ways we can help re-establish some degree of now-missing identity to your site, create a meta post and start talking about it on your site. Consider focusing your meta post on a specific part of the design - the background, the header content, the logo, etc. - as this will help limit the discussion so that it's not "everything is awful".
However at the moment this is for graduated sites only (see also this question on Meta Stack Exchange). As Catija mentions in her comment below this post there are plans to also allow this for beta sites, but apparently the discussion on this is still on-going.
Once beta sites are allowed to change their logo, we can hold a contest and select a winner. However I feel this has to be a decision of the entire Sustainable Living community. So if we go forward with this, we should create a post here on Meta where everyone can post their idea for a logo and we can simply use the voting system to see which logo is most popular.
There already was a brief attempt at this 4 years ago
but with little response. That is also the risk here. I think there is only a small group of people that visit Sustainable Living SE on a daily or weekly basis, and not many visit Sustainability Meta. If we have only a few people participating in this logo selection process chances are that the StackExchange community managers won't grant our wish for a logo.
To get more attention and have better and more proposed logos we could post a question in Graphic Design SE Meta to ask if people there are willing to help create logos.
For now we just have to wait....