Naming my water bottle?

Sharpie

Something like a koosie/coozie? ( spelling?) something stretchy but tight that you can draw, sew,.. on

Hello Mary, How about covering it with the leg end of a sock, to personalize it and embroider your name on that.

Add your initials using fingernail polish but test it on the bottom of the water bottle first.

Take it to a jeweler and have it etched/engraved. Decades ago I did that to a large thermos and gave it to my dad. (He took a lunch box and a thermos of iced tea every day to work)