Anchoring in Tile
On tile, 1/4-in. tubular plastic anchors hold firmly, whether there's drywall or cement backer board behind the tile. Be sure to buy tube-shaped anchors as shown here, not tapered anchors, which don't hold as well.
Enlarge the original anchor holes with a 1/4-in. carbide glass-and-tile bit. Tap in the plastic anchors and fasten the mounting plate with screws.