"Double strong" means it's been brewed twice (emphasis mine)
Like a strong-tasting cup of joe, but want more caffeine out of it? Start double brewing your coffee.
Though it sounds both simple and complex, double brewing your morning coffee doesn't require much beyond its name: just send your coffee through another brew cycle for a double-strong cup of joe.
It means a stronger taste and more caffeine, generally. There's nothing to suggest that O'Brien was ordering something different than this.
