1. I want some way to attach it to her walker while she's walking (I was thinking a strap with large finger-friendly snap).
Velcro! Velcro is much easier for elderly people with arthritis, muscle weakness, poor eye sight, etc (not that your grandmother has all or any of these). But, velcro is fast and easy.
2. It needs frills... that won't fray, will lay flat (to sit on) and are washable (I thought maybe some lacework using crochet thread).
-Place a ruffle around the edge. It won't be sat on/cause lumps (for this reason I would avoid cables ON the pillow)
-Use intarsia or fair isle to make a colored/decorative pillow rather than a one color pillow
-Make a two sided pillow. Two different colors/motifs so she can decide which side to sit on depending on her mood.