You can't install programs on an external hard disk.

You can only use it to store files, and even that isn't fully true, since external hard disks are notoriously unreliable, so it's best to say that you can only use them for backup and transport. If you really are going to use it for storage, you need a second one for backup, unless you backup to the cloud or a USB-stick, but the feasibility depends on how much data it is.

So the preferred option is to replace the hard disk by an bigger one, and then using an SSD in stead of a HDD is by far preferred. Go for a 256 or 512 GB SSD, and add an external hard disk of 1 or 2 TB for backup. You daughter wouldn't like to lose all her work, would she?

With an SSD the laptop will be noticeably faster.