1. Cook green beans in a large pot of boiling water just until tender-crisp, 7 to 8 minutes. Drain, cool in a bowl of ice water, drain again, and chill until ready to use.
2. Put potatoes in a large pot, cover with cold water, and bring to a boil over high heat. Reduce heat and simmer just until tender when pierced, about 15 minutes. Drain. When they're cool enough to handle, quarter potatoes and put them in a large bowl. Pour vinegar and olive oil over warm potatoes; add the onion, mix gently, and let stand until cool.
3. Meanwhile, make aioli: Finely shred peel of 1 lemon and squeeze enough lemons to get 7 tbsp. juice. In a food processor, whirl mayonnaise, lemon juice, mustard, garlic, 1/2 tsp. salt, and lots of pepper until smooth. Add lemon zest and pulse just to blend.
4. Mix cool potatoes with 1 cup aioli and the capers. Cover; chill until cold, along with remaining aioli.
5. Just before serving, gently stir in green beans, olives, more aioli if salad is too dry, and more salt and pepper to taste.
Note: Nutritional analysis is per serving.