I thought 祖母 was pronounced そぼ, so I was surprised to see ばば above the (祖母) Kanji in a quiz. I haven't found many good search results explaining this, the closest term I found was おばば.
祖母 can be read as そぼ, ばば, おおば, おば and うば. そぼ is the most common of them.
ばば is used as a term of endearment/familiarity with a grandmother or elderly woman. It appears in a few compounds too, like 祖父祖母 (じじばば), 曽祖母 (そうそぼ/ひいばば) and the 御祖母 (おばば) you found.
Written 婆, it can also be used to represent the Joker in card games, e.g. Old Maid = ババ抜き/婆抜き.
In case you were not aware, if you look up 祖母, そぼ, or ばば here in the dictionary it will bring up this, with all of the alternative spellings. You'll also see links to example sentences beside each version.
Similar to 祖母/ばば, grandfather has the reading 祖父/じじ (in addition to the normal 祖父/そふ) which is also used as a term of familiarity (like gramps, and gran instead of grandfather and grandmother).