Services and network features are the most important thing to consider when picking a host. Maybe even more so than hardware!
Minecraft servers are generally very competitive, so if your server is big enough, try looking for a host or hosting package with decent DDoS protection included as standard. Subdomains are generally provided for free, but some hosts take it one step further and provide a free dedicated IP on their more premium hosting tiers. This will allow you to get rid of the server port when giving other players your server's IP (by assigning it to Minecraft's default 25565).
It's worth looking for hosts where a test server IP is being provided. That way you'll be able to test for a lag-free connection and adequate bandwidth yourself.
Services are, in our opinion, what separates a good host from a great host. 24/7 and Config Support can be priceless when your very important server just starts crashing for no apparent reason in the middle of the night. Additionally, an Uptime SLA can guarantee your server will almost always be available!