pts = Partition[RandomReal[1, 10000], 2];
ListPlot[pts]

pts2 = {pts[[1]]};
Table[If[Min[Map[Norm[pts[[i]] - #] &, pts2]] > 0.05,
AppendTo[pts2, pts[[i]]]], {i, 2, Length[pts],
1}]; // AbsoluteTiming (* -> 1.35 *)
ListPlot[pts2]

pts = RandomReal[1, {10000, 2}];
f = Nearest[pts];

k[{}, r_] := r
k[ptsaux_, r_: {}] := Module[{x = RandomChoice[ptsaux]},
                      k[Complement[ptsaux, f[x, {Infinity, .05}]],  Append[r, x]]]

ListPlot@k[pts]

ops[pts_] := Module[{pts2},
  pts2 = {pts[[1]]};
  Table[If[Min[Map[Norm[pts[[i]] - #] &, pts2]] > 0.05,
    AppendTo[pts2, pts[[i]]]], {i, 2, Length[pts], 1}];
  pts2]

bobs[pts_] := Union[pts, SameTest -> (Norm[#1 - #2] < 0.05 &)]

belis[pts_] := Module[{f, k},
  f = Nearest[pts];
  k[{}, r_] := r;
  k[ptsaux_, r_: {}] := Module[{x = RandomChoice[ptsaux]},
                        k[Complement[ptsaux, f[x, {Infinity, .05}]], Append[r, x]]];
  k[pts]]


lens = {1000, 3000, 5000, 10000};
pts = RandomReal[1, {#, 2}] & /@ lens;
ls = First /@ {Timing[ops@#;], Timing[bobs@#;], Timing[belis@#;]} & /@  pts;
ListLogLinePlot[  MapThread[List, {ConstantArray[lens, 3], Transpose@ls}, 2],
               PlotLegends -> {"OP", "BOB", "BELI"}, Joined ->True]

pts = Partition[RandomReal[1, 10000], 2];

ListPlot[pts]

pts2 = Union[pts, SameTest -> (Norm[#1 - #2] < 0.05 &)];

Length[pts2]

326

ListPlot[pts2]

pts = Partition[RandomReal[1, 10000], 2];
nearestOnGrid[points_, d_] := Nearest[points, Outer[List, Range[0, 1, d], Range[0, 1, d]]~Flatten~1]~Flatten~1
testDistances[grid_, leastD_] := Min[EuclideanDistance @@@ grid~Subsets~{2}] < leastD

grid = nearestOnGrid[pts, 0.074]; // AbsoluteTiming
testDistances[grid, 0.05] // AbsoluteTiming
(* {0.000957, Null} *)
(* {0.016401, True} *)

GraphicsRow[{ListPlot[pts], ListPlot[grid]}, ImageSize -> 600]

result = NestWhile[
            Nest[ Complement[#, Rest@Nearest[ # , RandomChoice[#] ,
              { Infinity, .05}]] & , #, Ceiling[(Length@#)/100] ] &, pts,
                  Min[EuclideanDistance @@@ Nearest[#, #, 2]] < .05 & ];

pts = Last[NestWhile[{Complement[First[#],
      Nearest[First[#], x = RandomChoice[First[#]], {All, .1}]],
      Append[Last[#], x]} &, {RandomReal[1, {10000, 2}], {}},Length@First[‌​ #] != 0 &]];
ListPlot[pts]