Smooth/histogram a 2D set

Let's start with your sample data:

In[1]:= data = {{0.1, 1.0}, {0.2, 2.0}, {0.3, 3.0}, {0.35, 3.5}, {0.4, 4.0}, {0.5, 5.0}};

First we can use GatherBy to group entries by bin:

In[2]:= GatherBy[data, Ceiling[First[#], 0.2] &]

Out[2]= {{{0.1, 1.}, {0.2, 2.}}, {{0.3, 3.}, {0.35, 3.5}, {0.4, 4.}}, {{0.5, 5.}}}

Then select the second element of each pair (Last) and calculate the means:

In[3]:= Mean[Last /@ #] & /@ %

Out[3]= {1.5, 3.5, 5.}

I propose:

data = {{0.1, 1.0}, {0.2, 2.0}, {0.3, 3.0}, {0.35, 3.5}, {0.4, 4.0}, {0.5, 5.0}};

Mean /@ Reap[#2 ~Sow~ Ceiling[#, 0.2] & @@@ data][[2]]

{1.5, 3.5, 5.}

Another offering making use of BinLists:

data = {{0.1, 1.0}, {0.2, 2.0}, {0.3, 3.0}, {0.35, 3.5}, {0.4, 
    4.0}, {0.5, 5.0}};

Smoothed[data_, binSize_] := 
 With[{xs = data\[Transpose] // First, ys = data\[Transpose] // Last},
   Mean@ys[[#]] & /@ (Flatten[Map[Position[xs, #] &, #]] & /@ 
     BinLists[xs, 0.2 + $MachineEpsilon])]

Smoothed[data, 0.2]

(* ->  {1.5, 3.5, 5.}  *)