I don't think you can do this easily in Illustrator unless there's some plugin for this. I don't know how, frankly.
One "hack" I used in the past for similar work was in QuarkXpress (similar to InDesign, which you probably have too) and I didn't have the plugin to merge my data:
What I did was to import the "base" of design in Quark (for you, InDesign) and made a master page for the text and the design that didn't need to change.
On that master, I made a small text box that was linked to the text box of the next page; but I did it small enough to only be able to welcome ONE cell from Excel when importing the data. So what happened was that each cell containing a unique name had its own text box, on each page... and they were automatically added to each page. I hope I'm explaining this properly and I'm quite sure it's feasible in InDesign as well. You can also use a simple text document where each name is on its line or a page break; when you'll import that text in InDesign with the technique above, each line will go on a separate page, in the linked box where you want it to be.
That's how I did my hundreds of pages without having to copy/paste each name or use any extra plugin.
PS: Another benefit from do these invitations in InDesign is that it will be way faster for you to prepare the print-ready files! You'll be able to export in a few clicks all your invitations at once in a PDF; in Illustrator, you'll need to prepare each invitation individually!
Here's a quick visual how-to to hack-data-merge in Indesign:
I added more info in case it's your first time with Adobe InDesign!
1) Create a new document and select "primary text frame". Create the document the size of your invitation.
2) Create a Master Page and insert your Illustrator base layout. The pale blue zone represents your Illustrator image.
Note: By default, the "primary text frame" is almost the size of the page. On my screenshot it's smaller because I resized it.
3) Import your text in InDesign
It's possible you'll need to export your Excel file to text. If so, you should convert it to have pagebreak or each name on a separate line to make sure they will go in a separate text box!
If 2 names or more appear in the small text box, double-click again on the master page and resize it smaller vertically OR increase the text size!
4) Finalize the formatting and adjust size of the text box.