Unfortunately, the browser prompt can't be customized. But to attract a user's attention you can add a 'Push signup' template using Composer. The main idea of a Push signup template is to enroll a user in a push campaign with a remarkable call to action. If you add a Push signup template with the Composer card, the browser prompt will appear right after the Push signup template is displayed.
Below you can find instruction on how to add the Push signup template card to the Composer:
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Create Composer experience and add the Show push signup action card:
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Attach your push list to the Show push signup action card. Notifications will be shown to users from this push list:
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Create or attach an existing 'Push signup' template from the Management + Billing template library:
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Customize the Push signup template at your discretion: