Numbering Hierarchical Outlines

Python 2, 77

S={'0':0}
for w in input().split('\n'):S[w]+=1;S[' '+w]=0;print w[:-1]+`S[w]`

Like Sp3000's answer, but with a dictionary. The dict S stores the current number for each nesting level '0', ' 0', ' 0' and so on. For each line in the input, increment the corresponding nesting level, and reset the nesting level one higher to 0.

Python 2, 86 85 81 bytes

S=[]
for r in input().split("\n"):S=([0]+S)[-len(r):];S[0]+=1;print r[:-1]+`S[0]`

(-5 bytes thanks to @xnor)

Takes input as a string via STDIN, e.g.

'0\n 0\n  0\n  0\n 0\n0\n 0\n 0\n 0\n  0\n0\n 0\n  0\n  0'

Alternatively, here's a function for 5 extra bytes:

def f(I,S=[]):
 for r in I.split("\n"):S=([0]+S)[-len(r):];S[0]+=1;print r[:-1]+`S[0]`

CJam, 25 bytes

LqN/{0+I,<))_IW@toNo+}fI;

Like my Python answer, this uses an array to store which number each indentation level is up to. One difference, however, is that this uses t (array set) to replace the 0 on each line with the number we want.

Try it online.