How can I tell my device to ignore a public Wi-Fi network? The city I live in has city-wide Wi-Fi available for a subscription, but the Wi-Fi access point is public.

This means that everywhere I go in the city I get a notice that there is public Wi-Fi available even though I can't use it because I don't have an account with the city-wide Wi-Fi service.

How can I tell my phone (Droid X, Android 2.2.1, not rooted) to ignore Wi-Fi named "CityWifi"?

I'd like to get notification for other Wi-Fi networks, just not "CityWifi"
Try WiFi Ruler. 

I just downloaded it to see if it would work and it may be what you are looking for.  The app allows you to categorize known access points and then you can set rules on if your device will try to connect to that access point.  You could categorize "CityWifi" with one of the built-in rules to not connect.  And then all others you know of make it automatically connect and categorize as open or drive-by.  In the settings you can be notified of when you have connected to an AP.  My guess is that you would still have to turn off notifications for open access points in your settings like Matthew's answer.

Note: I am not the developer of this app