It can be run on Turris (2GB RAM), a full-featured web server (with graphics, comments, captcha …)
that would be for the occasional access? Per annum to 1000 hit?
Probably without problem. What means graphics?
My wife has some own web on http://www.webmium.com :), there are some pictures, forms, link to maps, menu… so graphics :).
And ASP.Net Mono + some db server? 
The configuration interfaces of Omnia use lighttpd as a webserver. If you can run your app as a CGI (FCGI, …) application, there’s no problem in running it on Omnia. Alternatively you can install nginx and use that for your website or, to isolate the service completely, run it in a container. That way most of the configuration should be same as on any other Linux server.
The resources should be sufficient for a website with low to medium traffic (all in all, some people run their Wordpress blogs on RasPi, which has way more limited resources), but we haven’t done any webserver benchmarks yet.
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