You could create a pseudo drop down menu using a popover containing a portal with the options. You can include a picture in a portal row. This is not possible to do in a native Filemaker drop-down menu.
You can also just show a portal with the same contents on the page and skip the popover.
The contents of the portal would have to come from a related table. That table would have to contain the relevant data, one of which would be a container storing the logo you want to display.
You ok creating relationships in Filemaker?
Be careful, or you will run into problems due to the "no portals may nest inside another portal" limitation.
If the kids have a portal listing each business and clicking on a row selects the business and then displays a popover, that popover cannot then contain a portal as it would nest one portal inside another and FileMaker won't let you do that.
You can, however, use a repeating field to list a small number of images for selection in that popover. Or you can place the popover button outside the portal of companies and use a regular button in the portal row to:
Set a match field in the layout's current record that enables the popover button's portal to show the correct set of image records.
Uses go to object to open that popover now that the portal can display the right images.
You can make the actual popover button invisible with a Hide Object When expresion of: True -- which will always hide the button when in browse mode and yet go to object can still open the popover button's panel.