With the help of Wim Decorte, I have figured this out. Admittedly, my understanding of the way ODBC works is very limited, but I will make an attempt here to explain it in hope that someone else will find it useful.
The problem is that there are 64 bit and 32 bit ODBC data sources and FileMaker can only see and use the 32 bit types. By default the 64 bit versions of Windows 7, Windows Vista, and Windows XP create 64 bit data sources when using the ODBC Data Sources control panel. So far so good, that makes sense. It does get a bit confusing from here, however.
Creating 32 bit data sources in a Windows 64 bit environment, requires the use of the 32 bit version of the ODBC data sources control panel. It is called odbcad32.exe and is located in the SysWOW64 directory. The 64 bit control panel is also called odbcad32.exe, however, and it is in, you guessed it, the System32 folder. Huh?!
So here is where we stand: There are two types of ODBC data sources: 32 bit and a 64 bit. There is nothing in the user interface of the Data Sources control panel to inform the user of this. The separate control panel applications for both 32 and 64 bit drivers have the same name, both ending in 32, and appear no different when in use. Pouring a fair amount of salt in the wound of confusion, the 64 bit driver is in a folder that ends with 32 and the 32 bit driver is in a folder that ends with 64. I don't mean to rant, but come on.
Well, what is a person to do. When I create a new System DSN in the 32 bit control panel, FM can see it. Ah, success at last!
Thanks againg to Wim. Hope this helps.