Well, if we were really starting from scratch, I would go with a base 16 numbering system (already in use by programmers everywhere).
One would have to throw out SI units, which would cease to make sense. It would also give us an opportunity to throw out the base 360/60/60 system (degrees/minutes/seconds) and 24/60/60 (hours/minutes/seconds) that the Babylonians saddled us with.
The proposal: The sixteen digits 0123456789abcdef. Integers would be written
with four digit groups (16 bits each). Mixed numbers would be witten
and a decimal separating fractions. Floating-point numbers (or numbers in scientific notation) would always be written
where "xPy" means "x × 16y" (P for exPonent.)
The new minute would be 1/256 th of an average day. (5.625 old-minutes) The new second would be 1/ 65536 th of a day (1.318 old-seconds).
(Note that you could always specify meeting times with only new-minutes, since one never wants to be more specific than 1 new-minute for such things. Therefore, "meet me at 1:15 p.m." becomes "meet me at :8D". Days and time can be specified simpler: "2007-10-31 13:46:45" becomes "7D7-A-1F:930" (Year 7D7, month A, day 1F, minute 93, second 0.))
The new standard length would be around 9.202 centimeters. that would make the speed of light equal to an even 16^8 (new standard lengths/new seconds)
The new degree would simply be 1/256th of a circle.
Translating to and from binary would of course be trivial.
One would have to come up with new prefixes which mean "256 times", "1/256 times", "1/65536 times", and "65536 times"—like "kilo" means "1000 times".