Meshing a cylindrical geometry with a notch

The issue can be addressed by specifying explicit numerical bounds:

<< NDSolve`FEM`
reg=RegionDifference[
  Cylinder[{{0, 0, 0}, {0, 0, 2}}, 1/4],
  Cuboid[{-0.25, -0.1, 0.9}, {-0.15, 0.1, 1.1}]];
RegionPlot3D[reg,PlotPoints->100]
ToElementMesh[reg, {{-1/4,1/4}, {-1/4, 1/4},{0,2}}]

Here is an alternative with exact numbers:

r = RegionDifference[Cylinder[{{0, 0, 0}, {0, 0, 2}}, 1/4], 
   Cuboid[{-1/4, -1/10, 9/10}, {-15/100, 1/10, 11/10}]];
m = ToElementMesh[r, MaxCellMeasure -> {"Volume" -> 0.001}, 
  "MeshOrder" -> 1]

But since you want a fist order mesh you could also discretize the regions first and then construct the region difference. This would allow for a good intersection of the regions:

r = RegionDifference[
   DiscretizeRegion@Cylinder[{{0, 0, 0}, {0, 0, 2}}, 1/4], 
   DiscretizeRegion@
    Cuboid[{-1/4, -1/10, 9/10}, {-15/100, 1/10, 11/10}]];
m = ToElementMesh[r, MaxCellMeasure -> {"Volume" -> 0.001}, 
  "MeshOrder" -> 1]

m["Wireframe"[PlotRange -> {8/10, 12/10}]]

This will requite a few more elements but it might be worth it.