there are a couple of ways to approach this, if you’re comfortable with php, you can just use php’s date functions to print directly to/within the page/box. you could also get more complicated (which isn’t really needed, since the php date functions are pretty simple to learn if you’re not familiar with them) and use a perl or shell script as a cron job, and the date functions to figure out what month/season then write to either the Db or a text file, then have your page read whichever of those you choose. (a system like that is more suited for something like the random quotes, etc type things which would be updated/changed more frequently).
have a look at the plugins area via the “extend” link above, i think there also may be a plugin or two available that either acomplish this already, or can easily be slightly modified to do it (quotes/random quotes plugins, etc).
if you’re really feeling adventureous, you could write a simple plugin using php (instructions in the wp codex plugins section) to do it.
