Cover the tiles surrounding the one you need to redo with masking tape to avoid scratching them in the process.
Remove the grout from the joints around the tile using a rotary tool. Place a carbide grout blade onto the tool then run the blade up and down the grout line, cutting out the grout until your reach the countertop core beneath the tile surface. If the joint on one or more sides of the tile are filled with caulk instead of grout, then cut the caulk into 3-inch strips with a utility knife and then strip it out from the joints with the edge of a putty knife.
Take a pry bar and slide it through one of the cleared joints surrounding the tile and under the tile. Apply pressure to the bottom of the tile to pop it out of its place in the countertop.
Cut the adhesive remaining in the countertop hole free with the rotary tool and carbide grout blade. Brush out the adhesive with a nylon brush. Sand the rear of the tile with coarse sandpaper to remove any trace of adhesive from its surface. Wipe clean with a piece of cloth.
Butter the rear of the tile with thinset mortar, using a notched trowel. Place the tile into the cleared hole, making sure that you position the tile squarely with the surrounding tiles. Place a carpenter’s level over the tile and adjacent tiles to make certain the tile is level. Add more mortar to raise the level of the tile, or remove some mortar to lower the tile level. Wait 4 hours for the mortar to dry.
Use a grout float to place grout into the joints surrounding the tile. Place caulk instead of grout into joints lying adjacent to moisture-rich surfaces. Remove any grout from the tile within 15 minutes with a damp sponge to prevent its drying in place, then wipe with a clean cloth 2 hours later to get the rest of the grout residue.
Allow the grout to dry for about 10 days then brush the grout with a tile and grout sealer to add a bit of stain-resistant waterproofing to the grout. Wait two more days before using the tile again.