My wife has this growth on top of her head which looks as if its brain tissue protruding from a tiny hole in her skull. What is this?

Mole on scalp. There are some lesions that can be benign that look cerebriform (brain texture-like with folds and crypts) such as moles (nevi), warts (verruca) and seborrheic keratoses. As this is a new lesion, it should be evaluated in-person by a dermatologist. Often biopsies are suggested on new lesions to be sure they are benign.
Have it checked! If your wife had (from birth/or very young age) a mole in the exact area and then the mole changed after puberty becoming more raised and brain like then she may have a nevus sebaceous. If this growth has more recently changed further i would be worried about it undergoing a malignant transformation. Have it checked by a dermatologist.
Granulation tissue. Sounds like granulation tissue from a wound; small wound in the scalp gets infected and the tissues around it go into overdrive giving a dark strawberry looking material. This usualy requires excision and treatment through your local doctor. Definitely not brain tissue though so rest easy; not dangerous.