This has been a strange interest of mine - when I first started watching game shows in the 1960s, virtually every show used nameplates to identify the contestants (Jeopardy, Eye Guess, etc.) or they didn't use anything at all (Chuck Barris' shows). The first show I remember using name tags was The New Price Is Right (for logistical reasons, as far as I know - people who have been in the audience can explain that better); the first one I remember using name tags regularly just for identification was Family Feud (probably because with ten contestants on stage and moving around as well as a relatively inexperienced host, it was almost a necessity). Now many shows use them (Pyramid and The Newlywed Game seem to be the holdouts).
My guesses as to the reason for the changeover are:
1) Most games aren't like Jeopardy where the contestant doesn't leave his/her podium after the introduction - they move around, so it's easier for the host to see the name accompanying the contestant.
2) It became cheaper and quicker to make a name tag than it is to have a stagehand set up the name on the podium.
3) Keepsake for the contestants.
Thoughts and corrections welcome.