Adding dynamic text to layout from attribute table with Data Driven Pages

The only attribute table accessible through Dynamic Text is the attribute table of your Data Driven Pages index layer. I'm assuming your Data Driven Pages map book is based off of your grid layer, and if so then you're in luck.

Using a dyn tag of type 'page', you can access any property of the index layer

<dyn type="page" property="YOUR_ATTRIBUTE_HERE"/>

For example, to grab the 'GridName' of the current page, use:

<dyn type="page" property="GridName"/>

You have probably looked at Using Dynamic Text with Data Driven Pages in the ArcGIS help files, but you may have missed the 'Tip:' section near the end of the page that explains this.