Open the Part and go to File > Properties. That will open a table similar to the one shown below. Type "Weight" (or whatever you want) in the Property Name cell, then select "Mass" from the drop-down in the Value / Text Expression cell. The value will automatically appear in the Evaluated Value cell (assuming you've correctly assigned the Material to the Part). Click OK.
Next, open the Drawing, with the drawing view inserted, and activate the Note function. With your cursor in the desired spot, click on the "Link to Property" icon in the note's property manager.
Select your custom property from the drop-down and click OK.