Most visitors to Makkah are there for Umrah, so the question is really what to do between prayers. The main options are a Ziyarat tour by private car to the historic sites around the city, the Revelation Exhibition and Holy Quran Museum at the foot of Jabal al-Noor, the Museum of the Prophet's Biography in the Clock Towers complex, and the Clock Tower museum itself. Each is a short trip rather than a full day, which is what makes them fit around prayer times.
Starting prices, confirmed before you pay. Tours we run ourselves are priced per vehicle for the whole group.

Makkah
Visit the historic sites around Makkah with a private car and knowledgeable driver.
from 170 SARper vehicle · about 43 SAR each for 4

Makkah
A guided journey through the Prophet's life and Islamic civilisation at this landmark museum in the Clock Towers complex.
from 70 SARper person

Makkah
Ascend the Makkah Clock Royal Tower for exhibits on the giant clock, astronomy and timekeeping, plus balcony views.
from 150 SARper person

Makkah
Visit the Revelation Exhibition and Holy Quran Museum at the foot of Jabal al-Noor, site of the Cave of Hira.
from 25 SARper person
The museums in the Clock Towers complex are walkable from most Haram-area hotels. Everything else — the Ziyarat sites, Jabal al-Noor, and the day trip to Taif — needs a vehicle, and around prayer times traffic in central Makkah moves slowly. We price Ziyarat as a private car for the whole group rather than per person, so the cost doesn't change whether two of you go or six.
Ziyarat private car in Makkah