The query critera request a name or text on the open dialog box. Is there any way to have the open dialog box show a dropdown list from that field to select the name?

Use a custom built form with a combo box that has the criteria values you want, just set the query criteria to that form and control.
HTH

How can I do the same with a report?

Base the report on the query.

I guess I'm not understanding the report thing. I now have a report that is based on a query. When I open the report, a dialog box pops up requesting me to type in a text. On this one it would be a name from a list of 900 different names.
What my user wants to do is when he opens the report he wants to click on a name to set the criteria for the query to run the report instead of typing in the name. Many times he has misspelled it or had too many spaces. How can I do this?

Make sure your query is running properly off of the form first. Then tie your report to it. Since it's prompting you for the text again, my instinct says you made a field name error somewhere in your report.

HTH,
David R