The premium item on the Re-Tail sign reset

This feature was not included in New Leaf. Instead, they have a shop called Re-Tail which is essentially a flea market where you can put your own items for your villagers and friends to purchase.

Villagers will infrequently ask the player for furniture that match certain size-based criteria. Sometimes they will ask for situational pieces like your instance, and in other cases they will ask for items that take up as much space as a random piece of furniture that they currently own. (For example, I've been asked to find an item that takes up as much space as a Pyramid before... 4 units of flooring in a square shape.)

Judging from the context of your villager's request, they are very likely looking for a furniture piece that takes up one unit of flooring. Checking the size of an item is pretty straightforward: simply take it home and place it in your room! Items that take up 'one unit of flooring' will appear right in front of the player -- larger items that take up two or more units of flooring will move away from the player to fill the appropriate space.

As far as I know, villagers won't turn down any pieces of furniture that fill the space requirement they are looking for. This is also true for bugs and fish, if you would like to unload any of those to quickly finish the task.