Your grandmother isn't the only one - I've seen people slip in bathrooms with tiles that look safe. What the other comments say about matte finishes and textured surfaces is fine, but there's a practical detail a lot of people miss: the size and shape of the tile matter a ton too. Small tiles (like mosaic or 20x20) with an anti-slip pattern work better than big smooth ones because they have more edges where your foot can grip, even if the texture wears down over time.
Now, about the maintenance they mentioned earlier, here's what nobody says: if you use a tile with really deep texture, dirt piles up in the crevices and then those damp corners become slippery again, almost like it's smooth again. The solution is to look for textures that are subtle but effective, not huge engravings. There are brands that specifically sell tiles with microsurface or cerealized finishes that look almost smooth but have real friction when wet, and they're pretty easy to clean.
A trick that actually works: test it with a wet tile by pressing with your finger or palm, don't just look at it. A lot of stores have samples and you can get your hand wet and check if it actually slips or not. It's silly but it saves you from surprises after spending money on the job. And if you want to be completely sure, there are adhesive anti-slip tapes you can stick on top of regular tiles - it's not the most aesthetic but it gives peace of mind while you figure out how to change things.