In any case if your DNS will be down, means that you HTTP server will also be down, because they are on the same server. So in any case user will not be able to see the webpage.
I have a dedicated server and my host has offered me the option of having the Domain Name Server for my sites on my server. This would save me having to request an entry from him for every new domain name I want to add - I could just do it myself. But I'm not clear what the implications of having this on my primary web server are?
You would need to keep the security patches up to date. It sounds to me like you are being offered a branded nameserver as in ns.yourdomainname.com but the secondary namserver will probably be the hosts's nameserver under an alias.
It would be possible to add and update zonefile entries through a web interface which would make things a bit easier for you (if you were comfortable with administering things like this) and for your host.
If you only administer a few domains that rarely change, then I don't think it is really necessary to have your own nameservers, unless you are chopping and changing on a frequent basis. At the risk of a very bad pun, it is all about delegation. :)
Regards...jmcc