In the general leaflet.js environment the attribution control is shown by default in the lower right corner. It combines attribution attributes from individual layers and is not easily manipulated.
As far as I know there is no way to modify the attribution settings via the {tmap} package.
If creating maps via the {leaflet} package directly (i.e. not via {tmap}) you can add a custom text to the little grey attribution box commonly displayed in the lower right corner by setting text in the attribution argument of addTiles() call.
For example including
addTiles(attribution = 'Eat mor chickin!')
Will lead to attribution along these lines:
Note that the attribution always starts with a link to leaflet, and continues with attribution of the tiles used (in this case OpenStreetMap).