Is it possible to have the Display name field automatically populated? Maybe put the Remote Computer name into the Display Name field if nothing is specified?

It already does auto-populate. If you click in the "Remote Computer" field while "Display Name" is empty and start typing, the "Display Name" will fill in automatically and match the "Remote Computer" field. If the "Display Name" and "Remote Computer" fields match and you change the "Remote Computer" field, the "Display Name" field will also follow those changes. If you put anything in the "Display Name" field (such as a space), it won't autopopulate because it is assuming you want a different name.

Does this answer your question / address your suggestion?

I'm open to different ideas with the "Display Name." :-)

In the upcoming 2.0 version of RD Tabs, there will be an icon in the tabs themselves to convey connection state (think IE 7), so the "Display Name" won't change (unless you want it to) when the connection state changes.