The government has tried to shift from $1 bills to $1 coins to save money, but the public doesn't seem to favor this move. I think the shift would be more successful if $1 bills were removed from circulation in favor of $1 coins while simultaneously putting the $2 bill into common circulation.
Since people seem to prefer bills, I think the $1 bills need to be removed for the $1 coins to be more widely used; if people don't have a choice, they'll have to use the coins. However, people don't want to carry lots of $1 coins either, hence the first part about getting the $2 bill into common usage -- people shouldn't get more than one $1 coin per transaction, and I think they could live with that.
I think if these two parts were done together, rather than just introducing $1 coins and hoping the public will embrace them, there would be a greater chance for success, and the government would be able to save money since coins typically last longer than bills.