The problem is not in the licensing or action of the export function. It is a matter of understanding how the Export Raster function works.
Based on your description, you are trying to select some number of cells from the raster, then do as you would a standard feature class in ArcMap and Export using a right click on the Layer Name.
If you did this on a Featur eclass with features selected, the window that comes up would look like this:
By default, the "Selected Features" option is used when there are features selected in the layer.
Rasters work somewhat differently when you export them. Here is the export window on a Raster, showing the Selected Graphics option grayed out, as you mentioned.
This Help document explains about Exporting a raster in ArcMap. In it, you see this section about Selected Graphics:
The Selected Graphics (Clipping) will clip and export the raster
dataset based on the selected graphics in the display; this option
will retain the data inside the graphic. Alternatively, you can check
the Clip Inside option to remove the data within the graphic and keep
the rest of the data.
What this is referring to, is not about selecting specific records from the attribute table that you want to export, but rather selecting a graphic in the Data Frame, that you want to clip your raster to, like the polygon below:
You can create Graphics with the Draw toolbar (more information here).
Right clicking to export the raster with the graphic selected, gives you the following option enabled in the window:
If you do want to Export Records from your Raster Attribute Table, you need to follow the procedure from this Help Document: Exporting a raster attribute table
A possible reason for this may be that the features have been converted to graphics. This will show the points, but will not allow editing. Can you select the points using the Select Elements arrow?
If so, these points are graphics. You can remove them or turn them off in the Annotation Groups tab. Right click on your data frame in the table of contents (called Layers in your example). Select Properties and go to the Annotation Groups tab. If there is an annotation group, you can remove it or turn it off.
Note, the above screen capture is from ArcGIS 10.0. I'm not sure if it's different in 9.3.
Best Answer
ArcMap is very picky about enabling the option to clip a data frame to "Outline of selected graphics." I don't remember where I read it, but here's the trick. Before you create the graphic element you want to use for clipping, right click on the data frame (while in Layout View) and choose "Focus Data Frame" (shown below).
The outline of the dataframe will have hash marks on it while it is "focused".
Then, draw the graphic element you want to use for clipping.
The clip data frame to "Outline of selected graphics" option is now available.
The data frame is now clipped.