As noted in a comment by neuhausr there is a documented solution for this problem on the GDAL site GDAL info page for ESRI Personal GeoDatabase, but the solution is only for Linux.
If like me you find yourself on a 64-bit Windows platform and have 64-bit MapServer/GDAL you might find yourself scratching your head, as yes you can get the same error :-(
msDrawMap(): Image handling error. Failed to draw layer named 'OGE_1M_surface_GeologicUnit'. msOGRFileOpen(): OGR error. Open failed for OGR connection in layer `OGE_1M_surface_GeologicUnit'. Unable to initialize ODBC connection to DSN for DRIVER=Microsoft Access Driver (*.mdb);DBQ=C:/WxS/ms/apps/published/BGS_OGE_Bedrock_and_Surface_Geology/data2/OGE.mdb, [Microsoft][ODBC Driver Manager] Data source name not found and no default driver specified
For me the solution was as follows:
Download/install the 64-bit version of the ODBC driver: Microsoft Access Database Engine 2010 Redistributable
Now in your MapServer map file you need to add the following two CONFIG options:
CONFIG "PGEO_DRIVER_TEMPLATE" "DRIVER=Microsoft Access Driver (*.mdb, *.accdb);DBQ=%s"
CONFIG "OGR_SKIP" "ODBC"
These config options are discussed in GDAL bug 5594 ~ Problem reading MDBs (64-bit)