Getting High Sierra installed on that Mac, and associated with your daughter's Apple ID (not yours), may be a bit of an "interesting" task.
Lion is so old that I don't believe that it can access the Support article How to download and install macOS - Apple Support, or the App Store link for High Sierra, within. You would probably need to use a modern Mac or PC to put the .DMG file containing the Sierra installer onto a USB flash drive, so you can transfer that installer to the old Mac. Then you'd want to format a spare external drive, install Sierra on it (by running the installer under Lion),, start up from the external drive, wipe the internal drive, install Sierra on it, and make sure that the Mac was associated with your daughter's Apple Account. Finally, you could boot off the internal Sierra installation, visit the Support article, and use the App Store link to get the High Sierra installer.
If you do all that, you could then install
The current version of Firefox requires Catalina or higher. Current versions of most of the other major third-party browsers require Monterey or higher. The current version of Safari requires Sequoia or higher. So as you can see, High Sierra is just hanging onto the edge of being useful for browsing the Web by its fingernails.