Which of these four parts of a modern car takes its name from the days of horse-drawn travel?

You are right if you've chosen "Dashboard".

What is the first meaning of 'dashboard'?

Initially, the word 'dashboard' meant a barrier of wood or leather fixed at the front of a horse-drawn carriage or sleigh to protect the driver from mud or other debris called "dashed up" (thrown up) by the horses' hooves. However, there was no additional function other than providing a convenient handhold for ascending into the driver's seat, or a small clip with which to secure the reins when not in use.

This information might be interesting for you:

In 19th century, when the first 'horseless carriages' were constructed with engines mounted beneath the driver such as the Daimler Stahlradwagen, the simple dashboard was retained to protect occupants from debris thrown up by the cars' front wheels. Later, as car design evolved to position the motor in front of the driver, the dashboard became a panel that protected vehicle occupants from the heat and oil of the engine. With gradually increasing mechanical complexity, this panel formed a convenient location for the placement of gauges and minor controls, and from this evolved the modern instrument panel, although retaining its archaic common name.