It's a good idea to place the plastic bag holding the steak and marinade on a rimmed baking sheet or in a baking dish, in case of any leaks.This recipe is one of our Better Basics: 10 New Takes on Family Favorites, see the others.
Whisk together oil, soy sauce, lemon juice, brown sugar, and garlic in a bowl. Place flank steak inside a gallon-size resealable plastic bag. Add marinade, press out excess air, and seal. Let stand, turning bag occasionally, at least 1 hour at room temperature, or up to 6 hours in refrigerator (if refrigerating, let come to room temperature before grilling).
Heat grill to medium-high. Remove meat from marinade, reserving marinade, and pat off excess with paper towels. Grill, uncovered, turning once, until a thermometer inserted into thickest part registers 130 degrees for medium-rare, about 5 minutes on each side. Transfer to a cutting board and let stand 10 minutes.
Meanwhile, transfer marinade to a small saucepan and bring to a boil. Remove from heat.
Slice meat against the grain and arrange on a platter. Arrange arugula on top; season with salt and pepper. Drizzle 1 to 2 tablespoons sauce over meat and arugula. Serve with remaining sauce and lemon wedges.