As a female Londoner, I have to say it is best to stick with 'black' cabs - the drivers are licensed and traceable - whereas just about anyone can set up as a minicab driver.
Fares for black cabs are also strictly metered - no cab in London is cheap, but you are unlikely to be ripped off (or attacked for that matter) by a licensed 'black cab' driver. Minicabs often work on a basis of negotiable fares. I have several friends who have had unpleasant experiences from taking rides in unlicensed cabs.
Just as a guide, the black cab fare from central London to my home just outside the city (about 5 or 6 miles) is usually £7 sterling. Bus fare is £1 any journey in central London, night buses are a bit more expensive and run all night, but only 1 every hour on some routes. You can tell them because they have N before the route number. Underground trains stop at about 12.30am most nights, and about 11.30pm on Sundays. London is not a 24 hour city!!!
As Donut says, let us know where in London you will be staying/airport and we will let you have some area-specific travel info.
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Time is an illusion. Lunchtime doubly so.