Your answer was missing mutate id! Here is the solution using dplyr packge only.
jj %>%
gather(variable, value, -(month:student)) %>%
unite(temp, student, variable) %>%
group_by(temp) %>%
mutate(id=1:n()) %>%
spread(temp, value)
# A tibble: 6 x 6
# month id Amy_A Amy_B Bob_A Bob_B
# * <int> <int> <dbl> <dbl> <dbl> <dbl>
# 1 1 1 9 6 3 5
# 2 1 4 8 5 5 3
# 3 2 2 7 7 2 4
# 4 2 5 6 6 6 1
# 5 3 3 6 8 1 6
# 6 3 6 9 7 5 5