I would not rely on the obscurity of mini_httpd to improve security.
As mentioned in SilverlightFox's answer, there is an outstanding vulnerability. I am surprised that there are not further issues - it could be that people just haven't tested.
To me, as an embedded systems pen-tester, mini_httpd is a red flag. If I see this in the headers, there will nearly always be a vulnerability somewhere else. This is of course anecdotal, but I have seen a very strong correlation.
I would stick with a more well-known and tested daemon.
lighttpd is still relatively lightweight, and they are responsive to security reports. It would be a good choice if you are hoping to host rich, interactive content.
yaSSL by wolfSSL is a security-driven HTTP daemon that is worth looking at. It is very lightweight, and they provide pay-for support if required.
nginx has also become feasible for larger embedded systems in recent years.