I would like to see EX: DX Weezing stay. It can give most Pokémon-ex a run for their money. Chances are, it won't.
Both ATM Rock and EX: FrLg Pidgeot need to stay rotated out. ATM Rock makes playing Rare Candy a risky decision, and Pidgeot makes the game too broken (1 Comp. Search every turn? Way too fast!). Many archetypes rely on Pidgeot and pretty much die without it. ATM Rock may have been more balanced if it only worked against the active monster, and Pidgeot may have been more balanced if you had to flip a coin to use its power.
I would also like to see Shining Rayquaza stay, although it will never happen.
If it had to be one card, though, it would be Professor Oak's Research. Without it, just about every deck loses the ability to set up consistently. Oak's Research, Copycat, and Rocket's Admin were nice cards since they allowed you to get rid of unwanted cards in your hand.
For now, Cleffa will have to do if there are no shuffle cards, although it is much too vulnerable...