Does it really matter? I think eMansipator has a valid point, it can get confusing if you don't know whether one is talking about the client software or the currency units.
I think it matters.
"Bitcoin" is easily recognizable as an alternative to "bitcoin" and it isn't necessary to use the nonstandard or unofficial variation of "BitCoin." What would be the reason for capital C?
Perhaps for related naming schemes these would be appropriate also then?
MoZilla FireFox
MicroSoft InterNet ExPlorer
AdoBe PhotoShop
GnuMeric
AbiWordAudaCity
RhythmBox
KonQueror