SetOptions for CoordinatesToolOptions for DateListPlot?

The option value is ignored because DateListPlot calls Graphics`DateListPlotDump`iDateListPlot which only uses Options for these items:

{PlotRange, AxesOrigin, GridLines, GridLinesStyle, Epilog, Prolog, Frame, Axes, Ticks,
FrameTicks, DateTicksFormat, DateFunction, DataRange, PlotRangePadding, PlotLegends,
PlotStyle, PlotMarkers, Joined, BaseStyle, LabelStyle, TargetUnits, Method}

The values of these options are modified as needed, then combined with explicit options after the latter are filtered with:

opts = FilterRules[opts, Options[If[caller === DateListLogPlot, ListLogPlot, ListPlot]]];

Therefore we can patch the broken definition with this replacement:

DateListPlot; (* preload; do not remove! *)

With[
  {dv := DownValues[Graphics`DateListPlotDump`iDateListPlot]},
  dv = dv /.
    (fr : FilterRules)[a : Graphics`DateListPlotDump`opts, b_Options] :> 
      fr[Join[a, Options @ Graphics`DateListPlotDump`caller], b]
];

Now:

SetOptions[DateListPlot, 
  CoordinatesToolOptions -> {"DisplayFunction" -> MapAt[DateString, 1]}];

DateListPlot[{{1, 1, 2, 3, 5, 8, 11}, {5, 8, 9, 6, 2, 4, 7}}, {2013, 1, 1}, 
 Joined -> True]