Spray grill basket with nonstick nonflammable cooking spray.
Preheat grill to medium-high heat.
In a large bowl, combine potatoes, oil, salt, and pepper.
Place potatoes in grill basket, and grill for 10 minutes or until potatoes are tender, turning once.
Remove potatoes from grill.
Cool potatoes, and cut into bite-size pieces.
In a large bowl, combine potatoes, mayonnaise, basil or dill, parsley, mustard, and lemon juice, stirring until potatoes are well coated.
Serve at room temperature or chilled.
Garnish with fresh basil or dill, if desired.
Note: If fingerling potatoes are unavailable, substitute small new potatoes.