Which Mendix version are you using?
In data grids before Mendix 9, you cannot add buttons. You can however add a button above the grid that doubles as the default action when someone clicks or double clicks a certain row in the grid. So to solve your problem:
An alternative would be using a list view and adding a button in there.
If you're using the beta data grid in Mendix 9, the above answer is invalid since you can actually use widgets in that one.
You can try to use a list-view instead, With some css you can let it look like a datagrid. Then you can add a button with class 'btn-link'
Datagrid does not support this option by default. As a workaround, you can use template grid. With few tweaks, you can make template grid look like a data grid and it provides features like pagination, action buttons just like a data grid.
Set the template grid to 1 column and number of rows = page size you want.
Now add a table inside template grid item with desired number of columns/attributes to show in the grid. In any table cell, you can place a button/link widget. When user clicks that button, you can set a microflow that will be called. Object/row will be passed to the microflow.
For heading of columns like a data grid, use a table above template grid widget. Then you will need some CSS tweaks to make it look like a data grid.