Heya Hello Weasyl Staff. You guys do have a rather stable site that offers a large amount of submission types that others don't offer, but it is still pretty quiet even after all this time. I've come back on occasions to check on it and post this and that. I hear it mention off hand every few months as well, but not too often. I think that weasyl could use some advertisement to reach out to more audience once more.
FurAffinity offers advertising, and I do know that other art sites are allowed to advertise on there: http://www.furaffinity.net/advertising.html
Sadly, I can't really offer suggesting other advertising opportunities that you can do without breaking wallet, as the furry sites are kind of the sites that promote helping creators while sites like DA's ad service is more focused towards large scale businesses and super expensive. The ads on FA do reach a large amount of people, though, and I'd suggest perhaps have a human artwork in the ad somewhere as well to try to appeal to diversity. So far reaching out to folks has been pretty reliant on word of mouth, and in all cases I see people comment how "weasyl is pretty deserted" which isn't the best word of mouth promotion.
I could also suggest creating a DeviantArt group focused around this art site to help promote Weasyl on DeviantARt, too, to reach out to the non-furry artists as well. Collect fanArt of your Mascot drawn by DeviantArtists to help promote both the website, the character, and urge more artists to draw the character too. It'd be more work than needed, so its a bit of a hassle of a suggestion. I didn't see a weasyl group, though. https://groups.deviantart.com