I can't believe I found a solution to this and no one from QBO has been able to come up with it.

My Situation: I had an estimate that was approved. We do progress billing. We have had 2 invoices already billed and paid against the approved estimate. We have 1 invoice due from the client yet to be paid. I upgraded to Plus to be able to track each project separately. When I followed QBO experts' directions and changed the customer dropdown to the project name the entire invoice cleared out. 

Here is the workaround I found:

1. Create your Project.
2. Now, go to Sales and click on Invoices.
3. Go to the invoice that you want to migrate to your project and click on the "▼" Down Arrow on the right hand side (next to Edit | Print | ▼ ). Select Duplicate
4. The duplicated invoice will show up in your invoices tab with the same customer name and Amount as the original but it will have a different invoice number and will be marked as unpaid.
5. In the duplicated invoice go to customer dropdown and click on your project name.
6. You will now have 2 invoices with the same Amounts listed in your Invoices tab. 
7. You will have to go back and mark your Duplicated invoice as paid manually.
8. To make everything match exactly, you can go into your Duplicated Invoice and change the invoice number to match the original invoice number.
9. You will now see that the duplicated invoice labeled as "Customer Name": "Project Name" instead of just "Customer Name" 
10. The duplicated invoice will be what is in your project under the projects tab.
11. Once everything is as it should be, you can either Void or Delete your original invoice that you were unable to migrate to Projects.

Doing this saved me from having to recreate my invoices line by line (we break down our quotes by activity on most of our contracts, so most of our estimates and invoices have 20+ lines in them.) Hope this helps someone...