No, sorry, I don't believe that "bits" or "bytes" are SI units. The SI unit system was set up to measure fundamental physical quantities, and bits and bytes are artificial constructs, not measurable physical dimensional entities.
Just because we use metric-like prefixes (eg. kilo, mega, etc), with them doesn't make them SI units... does calling 1 million dollars a "mega-buck" make the dollar an SI unit?
Later edit...
Sorry to upset anyone, but if we're quoting Wikipedia...
The article on SI units says that there are SEVEN basic SI units:
metre m length
kilogram kg mass
second s time
ampere A electric current
kelvin K thermodynamic temperature
candela cd luminous intensity
mole mol amount of substance
In addition, there are a number of non-SI units accepted for use with SI:
minute min time (multiple unit) 1 min = 60 s
hour h time (multiple unit) 1 h = 60 min = 3600 s
day d time (multiple unit) 1 d = 24 h = 1440 min = 86400 s
degree of arc ° angle (non unitary unit) 1° = (π/180) rad
minute of arc ′ angle (non unitary unit) 1′ = (1/60)° = (π/10800) rad
second of arc ″ angle (non unitary unit) 1″ = (1/60)′ = (1/3600)° = (π/648000) rad
square degree deg² or sq.deg. solid angle 1 deg² = (π/180) sr.
hectare ha area (simple decimal multiple unit) 1 ha = 100 a = 10000 m²
litre l or L volume (simple decimal multiple unit) 1 dm3 = 0.001 m3
tonne t mass (simple decimal multiple unit) 1 t = 103 kg = 1 Mg
Then there are even more which are mentioned which are rare or discouraged. None of them is the "bit".
I stand by my comment. The use of a prefix is just that...it doesn't give the subject of the prefix any special SI status.