When you say "iLogic Form", do you mean that you have created your own iLogic Form, and the 'Checked By' is one of the iProperties being represented within that, or do you just mean the default iProperties dialog? If you are referring to the default iProperties dialog, you won't be able to alter that to show a drop-down list. For one thing, an iProperty can't currently hold multiple values, like a user parameter can, so in order to have a multi-value drop down list to choose other names from, you would first have to create a text type multi-value user parameter to hold the names. Then in your custom iLogic Form, you can add this user parameter, which will allow you to select one of its names from a drop-down list. Once that is set-up there are a few ways to get the value of that parameter to update the value of your iProperty. Since this is a text type parameter, it won't have the 'export' checkbox by it in the Parameters dialog box, which would have made this much easier, so we will have to use an iLogic rule.
The next step would be to create the new 'local' (saved in the document) iLogic rule, which will be triggered to run each time the value of that user parameter changes, then it will copy the value of that user parameter to your iProperty. Assuming you named your user parameter "CheckedBy", the local iLogic rule could be as simple as this:
'CheckedBy is the name of a 'local' multi-value text type user parameter
iProperties.Value("Project", "Checked By") = CheckedBy
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Wesley Crihfield

(Not an Autodesk Employee)