Can I just buy an eSATA adapter bracket for a few dollars and plug it into a SATA connector on her motherboard?
Check the bios setup for AHCI mode, for the SATA port.
That's what's needed for "Hot Plug" to work correctly.
However, if the Windows installation was done while in IDE mode, then you'll need to either re-install Windows, or do some registry editing to allow successful boot-up (of the internal HD) while AHCI mode is enabled.
Note: many motherboards use a non-Intel controller for 2 of the SATA ports. In that case, enable AHCI mode for the non-Intel ports, and use one of those for eSATA. The Intel ports can be left in IDE mode for the internal HD.