There's an option to hide the version so it will display only nginx, but is there a way to hide that too so it will not show anything or change the header?
Like Apache, this is a quick edit to the source and recompile. From Calomel.org:
The Server: string is the header which is sent back to the client to tell them what type of http server you are running and possibly what version. This string is used by places like Alexia and Netcraft to collect statistics about how many and of what type of web server are live on the Internet. To support the author and statistics for Nginx we recommend keeping this string as is. But, for security you may not want people to know what you are running and you can change this in the source code. Edit the source file
src/http/ngx_http_header_filter_module.cat look at lines 48 and 49. You can change the String to anything you want.
## vi src/http/ngx_http_header_filter_module.c (lines 48 and 49)
static char ngx_http_server_string[] = "Server: MyDomain.com" CRLF;
static char ngx_http_server_full_string[] = "Server: MyDomain.com" CRLF;
March 2011 edit: Props to Flavius below for pointing out a new option, replacing Nginx's standard HttpHeadersModule with the forked HttpHeadersMoreModule. Recompiling the standard module is still the quick fix, and makes sense if you want to use the standard module and won't be changing the server string often. But if you want more than that, the HttpHeadersMoreModule is a strong project and lets you do all sorts of runtime black magic with your HTTP headers.