The quickest way would be to add two new fields to your point table and calculate an Xnew and Ynew field which takes the form of
from random import uniform
def shift(val, start=-1, end=1):
"""shift within the range +shifty and -shifty"""
jiggle = uniform(start, end)
return val + jiggle
Now if val was 10, then you would get numbers in the 9-11 range.
Just repeat for the X and Y coordinates.
The 'val' can be provided from an appropriate X or Y field or you could use !Shape.centroid.X! and !Shape.centroid.Y! instead.
The expression box is
shift(whateverYouUseForVal, start=-1, end=-5) # you can mix up the start and end ranges, they don't need to be equal
EDIT
I failed to state the obvious... add the new x,y data back into arc* and save it as a shapefile/feature class