"Most popular mountain in Antarctica" is a category error. Popularity implies accessibility, repeat visitation, and cultural familiarity—none of which apply to a continent with no permanent residents, no commercial tourism infrastructure on peaks, and fewer than 100,000 annual visitors total (mostly coastal).
That said, certain peaks have notoriety within mountaineering circles:
All require polar mountaineering expertise, specialized gear, permits (Antarctic Treaty System), and expedition-scale logistics. "Popular" doesn't apply—these are objective-driven undertakings for elite alpinists and scientists.
For non-climbers: scenic flights from Punta Arenas (Chile) or Cape Town (South Africa) offer aerial views of the Ellsworth and Transantarctic ranges. That's as close as "popular" gets.