2010-11: Edwin Minney played for the Berks County Bruins AA midget minor team in Pennsylvania and was invited to USA Hockey's Select 15 Player Development Camp in Rochester.
2011-12: Minney played midget hockey with the Washington Junior Capitals and represented the USA in the 2012 Youth Winter Olympic Games in Austria. He was 6-9-7 in 25 games for the Junior Capitals and had a 2.71 goals against and .902 save percentage. In four games at the Youth Olympics he was 2-1 with a 4.65 goals against and .833 save percentage. Minney was invited to USA Hockey's Select 16 Camp and was selected to play in the National Team Development Program in 2012-13.
2012-13: Minney skated for the USA NTDP's U17 team in in his first season in Ann Arbor and competed in the 2012 Four Nations Tournament in Switzerland, 2013 World Hockey Challenge and 2013 Five Nations Tournament in Finland. In 35 regular season games he was 9-20-3 with a 4.04 goals against and .890 save percentage. Minney was 3-0 with a 2.25 goals against and .912 save percentage as the USA finished first in Switzerland. He was 2-0 with a 2.14 goals against and .857 save percentage as the USA finished third in the WHC and went 1-0-1 with a 2.42 goals against and .896 save percentage as the USA again finished third at the Five Nations Tournament.
2013-14: Minney returned to Ann Arbor for his second season with the NTDP — sharing the goaltending duties for the program's U18 team with Blake Weyrick. He was selected to play in the 2013 USA Hockey/CCM All-American Prospects Game in September and in November committed to playing college hockey at Michigan State in 2014-15. Minney was ranked second amongst North American goaltenders in Central Scouting's mid-term rankings prior to the 2014 NHL Draft.