How can I change the image that is shown in SAP's main menu screen?

It is not possible to do this with GuiXT, but you can do it by changing SAP customizing tables:

Upload the new image, e.g. a .jpg file,  to SAP Web Repository using transaction SMW0. Then change the image file name in the SAP table SSM_CUST, table argument "SAP_START", using transaction SM31.