Some other notes...
The team was also going to be called the Steers at first, then the Rangers, and finally announced on 03-16-1960 that the team would be the Cowboys.
Don Meredith and Don Perkins were both signed before the franchise was granted, and predated the expansion draft of other teams' unprotected players.The Cowboys had to give up 3rd and 9th round draft choices in 1962 to the Bears and Colts as compensation since these teams had drafted these two players after Dallas signed them to individual service contracts.
Of the 36 players taken by the Cowboys in their expansion draft on 03-13-1960, only Frank Clarke (TE) and Jerry Tubbs (LB) would remain with team for a substantial amount of time. Techinically, these players were drafted by the Dallas Rangers as the team would change names 3 days later (see above).
Meredith could have started day one, but Landry protected him knowing things were going to be rough for the first couple of years. Eddie LeBaron instead got the reps and abuse the first couple of seasons at QB.
Bob Lilly was the very first collegiate draft pick of the Dallas Cowboys in 1961. He would go on to have a Hall of Fame career and became known as "Mr. Cowboy".
Sources:
Total Cowboys (1998 HarperCollins)
Wikipedia - History of the Dallas Cowboys
Wikipedia - Bob Lilly