RegionMember with some tolerance?

This (or an appropriately written variant) works:

myRegionMember[a_, b_] := With[{threshold = 10^-6},
  If[
   RegionDistance[a, b] < threshold,
   True,
   False
   ]
  ]

There's Internal`$EqualTolerance. See How to make the computer consider two numbers equal up to a certain precision and its reference.

Block[{Internal`$EqualTolerance = 9.},
 RegionMember[Line[{{0, 0}, {1, 10.^-7}}], {.5, 0}]
 ]
(* True *)

It's a relative tolerance, whereas gpap's & kguler's answers give absolute ones. The setting above, which is roughly equivalent to Internal`$EqualTolerance = $MachinePrecision - 7, says approximate numbers that agree to seven digits (or differ in at most the last nine) are to be considered equal.

Block[{Internal`$EqualTolerance = $MachinePrecision - 7},
 {1. + 1.0000001*^-7 == 1, 1. + 1*^-7 == 1}
 ]
(* {False, True} *)

f1 = RegionDistance[#, #2] < #3]&

f1[Line[{{0, 0}, {1, 10.^-7}}], {.5, 0}, 10^-6]
(* True *)

f2 = Chop[RegionDistance[#, #2] ,#3] == 0&

f2[Line[{{0, 0}, {1, 10.^-7}}], {.5, 0}, 10^-6]
(* True *)