OK, only reason I asked was because her risk factors had an almost "Can You Top This?" kind of vibe: 87, lives in a group home, one-shot J&J ...
Anyway, I want to echo what Alex said: I'm glad that you came here and asked the question. I also have parents in their 80s, and I know how stressful and confusing the past couple years have been. Wishing her good health.
By the way, I can't remember if this was posted in the thread when it came out back in September (presumably it was), but for anyone who missed it, I encourage you to read this
eye-opening article from NYMag about Covid and age:
Think of your 87-year-old mom with one shot as being like a 65-year-old unvaccinated woman. If she gets the booster, maybe her risk profile is more like that of a 50 year old unvaccinated woman: still not completely out of the woods, but way safer than before she was boosted.