Cheryl,
I'm at a loss as to what could possibly be causing what you describe. JFBConnect is used on a wide variety of hosts, including very cheap hosting plans with much, much less memory than you describe above.
My immediate suggestions would be to:
* Make sure Debug Mode in the Joomla Global Configuration area is off
* Make sure Enable Debug Mode in JFBConnect is set to "No"
* Set the Caching option in JFBConnect to 15 (or more)
If none of those help or were already configured that way:
* Disable any Social Profile plugins you have enabled and test again. If the spike occurs with them enabled, but goes away when not, try going through the settings in each one and determine if there is a specific setting causing the problem.
Beyond that, I'd recommend testing what the standard memory usage is across a variety of pages on your site. If you enable Debug Mode in the Global Configuration area and have the System - Debug plugin enabled, there should be a very nice output at the bottom of each page showing memory usage and database query logs. On normal pages, JFBConnect usually adds about 1MB and 4-6 db queries (which isn't much). We generally recommend trying to have the db query count under 100 and memory usage under 100MB.
I hope that helps get you started on narrowing things down, but please let us know what you find.
Thanks,
Alex