As one who loves Gopher Football history - allow me the luxury to go over a bit of background regarding the Minnesota v. Illinois rivalry (and likely why the game never had a trophy). Illinois was national champions in 1923 - Red Grange's sophomore year. In 1924 they came into Minneapolis for the Memorial Stadium dedication game with an excellent 5-0-1 record and were heavily favored. Gophers beat them 20-7, thus preventing Illinois back-to-back national championships (finished 2nd to the "Four Horsemen of Notre Dame"). Legendary Illinois coach Robert Zuppke was so incensed about this loss he supposedly vowed to never play the Gophers again. So no Gopher - Illinois games occurred until Zuppke's last season in 1941 (whereby Bierman's national championship team handled them rather easily). Hard to generate a rivalry (and trophy) when you're not playing each other.
FWIW - I think we have too many trophies as it is - three should be the max (jug, pig, axe). The more trophies one has, the more they are diminished in importance. Sort of like giving everyone "participation ribbons" when they play.