The historian and muhadith (narrator) Ibn Kathir writes the following:

Bakkah is one of the names of Makkah. Bakkah means, "it brings Buka'" (crying, weeping) to the tyrants and arrogant, meaning they cry and become humble in its vicinity. It was also said that Makkah was called Bakkah because people do Buka next to it, meaning they gather around it. There are many names for Makkah, such as Bakkah, Al-Bayt Al-`Atiq (the Ancient House), Al-Bayt Al-Haram (the Sacred House), Al-Balad Al-Amin (the City of Safety) and Al-Ma'mun (Security). Makkah's names include Umm Rahm (Mother of Mercy), Umm Al-Qura (Mother of the Towns), Salah, as well as others...

p.s. As you can imagine, this is badly translated from the original, but you get the idea. Here's the source.

p.p.s. If you're interested in the implication from a theological point of view: Is It Makkah or Bakkah?