I have a number of websites hosted on a dedicated server running Linux. Is there any specific things that I should do on my server to mitigate/prevent any impact from a DDoS against one of the site I host?
Nothing within your power can protect you from a multi-gigabit DDoS attack except bandwidth or a black hole route.
If you're getting smaller attacks you can try Fail2ban, some Nginx rules, or an edge cache.