The actual way to spell "Lion" in Swahili is "Nsimba" not "Simba". For the movie "The Lion King", they took off the first letter to make it easier to pronounce for American audiences (and anyone whom isn't African or doesn't fluently speak Swahili).
In Shona Simba means power, the most common full form of the name is Simbarashe, 'rashe' means 'of God' therefore the English translation would be power of God. Otherwise in Shona again the root name Simba could be in the form of Masimba meaning 'Ma' meaning 'much' therefore the full name would be translated to mean 'much power'.