Grandma Bottle is Bert's grandmother. In the short story "Bedtime Story," from The Sesame Street Library Volume 13, Bert and Ernie are staying at Grandma Bottle's house. They share a bedroom whose wallpaper indicates Grandma's penchant for bright yellow and splotchy flower designs. The room decor also features a framed portrait of Bert holding a slate, presumably a moment which Grandma treasures.
Little is known about Grandma Bottle's personality, or whether her name influenced Bert's passion for bottlecaps, but she does expect her guests to make their own beds.