How to delete cell contents in Word with VBA?

This works:

ActiveDocument.Tables(1).Cell(1, 2).Select
Selection.Delete

This deletes the cell contents but leaves the empty cell behind.

I understand your dismay, because oddly, the above does not do the same as

ActiveDocument.Tables(1).Cell(1, 2).Delete

which deletes the entire cell!

The former is the equivalent of selecting a cell and pressing the Delete key (which clears the contents but leaves the cell in place). The latter is the equivalent of right-clicking a cell and choosing "Delete cells..." (which deletes the cell).