GMT20161003-randomtext.mp4

zmodload zsh/datetime
autoload zmv
zmv -n '(**/)GMT(<->)*(.mp4)(#qD.)' '$1${(L)$(
  strftime %A "$(strftime -r %Y%m%d $2)")}$3'

find . -name 'GMT*-*.mp4' -type f -exec bash -c '
  for file do
    base=${file##*/}
    date=${base#GMT}
    date=${date%%-*}
    wday=$(date -d "$date" +%A)
    echo mv -i "$file" "${file%/*}/${wday,,}.mp4"
  done' bash {} +