Thanks for your posting. The dropdown for previously entered values is a feature provided by the browser and this is not available for all fields (as they are technical differences between the Version field and Name for test runs, for example). We will look into adding direct support to TestRail for this feature to a future version.
It would alternatively be possible to add a custom dropdown box to the form as well (in the administration area under Administration > Customizations).
I hope this helps. Just let me know in case you have any further questions.