<$button set="$:/themes/tiddlywiki/vanilla/settings/backgroundimage" setTo="FirstImageTiddler">
Image 1
</$button>
<$button set="$:/themes/tiddlywiki/vanilla/settings/backgroundimage" setTo="SecondImageTiddler">
Image 2
</$button>
If you want an action widget instead because you want to be able to do multiple actions on the same widget, you could instead use an $action-setfield widget, like:
\define setAction(imageName) <$action-setfield $tiddler="$:/themes/tiddlywiki/vanilla/settings/backgroundimage" text="""$imageName$"""/>
<$button actions=<<setAction "FirstImageTiddler">>>
Image 1
</$button>