If you would like to use the EquationNumbered style for typesetting, but using your own equation numbers or designations instead of the automatically generated ones, you can simply edit the cell expression by going into the menu Cell > Show Expression. If you do this on a blank EquationNumbered cell, you should get code like the following:

Cell[BoxData[ FormBox["", TraditionalForm]], "EquationNumbered"]

There, you need to override the CellFrameLabels option of the parent Cell, with the following code:

{{None, Cell[ TextData[{"(YOUR_DESIGNATION_HERE)"}], "EquationNumbered"]}, {None, None}}

Substitute the YOUR_DESIGNATION_HERE text with the number or designation you want to assign to your equation. For example if you want to call it A1, your cell should end up like this:

Cell[BoxData[ FormBox["", TraditionalForm]], "EquationNumbered", CellFrameLabels->{{None, Cell[ TextData[{"(A1)"}], "EquationNumbered"]}, {None, None}}]

Now you can disable the menu Cell > Show Expression option, and your cell is ready for you to type your equation in!