If you open Project Detail Page called Project Information, you can see default field for Owner. It is a people picker field:
Well, if this field type exists on default Project Server Pages, how come it can not be added to customized page?
I actually don't know the answer to that question, but it is a fact that you can not add this field type to your page.
But, there is another solution. People picker for Project Server exists in another form. In some of my previous posts, I mentioned Solution Starters package from Microsoft (when I talked about Dynamic Workflow).
In this package, you can find solution for people picker field, it is called PDPCustomization.
If you want to use people picker field in your forms, first you need to download Solution Starters package from this link:
http://archive.msdn.microsoft.com/P2010SolutionStarter
EDIT (8.6.2014.)
Microsoft has retired link for downloading this solution, so for downloading Solution Starters go to these links depending on the version of Project Server you are using:
Project Server 2010:
https://drive.google.com/file/d/0Bz62Vp3ZwGCrRDBCdjNrYUNYWmM/edit?usp=sharing
Project Server 2013:
http://www.fluentpro.com/solutionstarters2013.html
Look for PDPCustomization.zip file in that package, extract it and deploy solution to your Project Server.
After the solution is deployed, to add a people picker to the page, web part of deployed solution must be added to the page. Web Part is called AutoComplete Web Part and it is located in PDP Customization folder:
Position this new AutoComplete Web Part under the existing Web Part on the page.
Click on Edit Web Part on the Web Part menu. Menu with properties of that Web Part will open. In that menu, you can choose the group from which users will be fetched and field name in which the name of this user will be saved (this field is Custom Field of type "Text").
After the changes have been saved, your custom field will now read users from Administrators group.