Hello Carsten Koster,
I'll share with the community my response to your support ticket here, as well:
We have an exposed configuration for localizing the resources and since the SearchBox is using a resource for the text that is displayed in the Button, you can localize it from Administration >> Interface Labels and Messages and there look for this resource:
Used in : SearchResources , Resource key : Search
When you click on it, you will be able to specify different labels for all the frontend cultures that you currently have in your project.
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