Let's say that your would like a Grid Bubble to show both the grid name and something else. Here's how to do it...
You will need to make a custom Grid Head family with the lines and circles desired using the grid family template.
You can add two labels, the Grid Number and a custom parameter. You will also have to make a ‘Shared Parameter’ for year or phase, or both.
Then you will load that family into the project and apply it to the System Grid family.
You will also need to add the Shared Parameter to the Project Parameters. Then you will be able to type the year or phase into the grid.