I don't know if this is 'the' coconut cake a friend of mine was looking for, but found this on Paula Deen Recipes.com. Here's a link to the website:

Jamie's Coconut Cake


1 cup (2 sticks) butter, at room temperature
2 cups sugar
4 eggs
3 cups sifted self-rising flour
1 cup coconut milk
1 teaspoon pure vanilla extract

Preheat oven to 350 degrees F.

Grease and flour 3 (9-inch) cake pans. Using an electric mixer, cream butter until fluffy. Add sugar and continue to cream well for 6 to 8 minutes. Add eggs, 1 at a time, beating well after each addition. Add flour and milk alternately to creamed mixture, beginning and ending with flour. Add vanilla and continue to beat until just mixed. Divide batter equally among prepared pans. Level batter in each pan by holding pan 3 or 4 inches above counter, then dropping it flat onto counter. Do this several times to release air bubbles and assure you of a more level cake. Bake for 25 to 30 minutes or until done. Cool in pans 5 to 10 minutes. Invert cakes onto cooling racks. Cool completely and spread cake layers with your favorite frosting to make a 3-layer cake.


Filling:
3/4 cup sugar
1 cup sour cream
4 tablespoons milk
1/2 cup flaked, sweetened coconut

Frosting:
7 Minute Frosting recipe
Flaked, sweetened coconut for sprinkling

Directions
Follow directions for Basic 1-2-3-4 Cake, substituting coconut milk for regular milk. While cake is baking, prepare filling. Stir together sugar, sour cream, milk and coconut in a bowl until well blended. Remove cake layers from oven and allow cake to remain in pan as you prepare to stack and fill. Remove first layer and invert onto cake plate. Using the wrong end of a wooden spoon, poke holes approximately 1 inch apart until entire cake has been poked. Spread one third of filling mixture on cake layer. Top with second layer, repeat process. Top with last layer and repeat process again. (As I stack layers together, I stick them with toothpicks to prevent cake from shifting.) You can place the cake in the refrigerator for 3 days to allow it to absorb the filling mixture.

Prepare 7-Minute Frosting. Frost top and sides of cake. Sprinkle top and sides of cake with additional coconut.

Servings: 1 three-layer cake
Prep Time: 30 min
Cook Time: 25 min
Difficulty: Easy

7-Minute Frosting

1/3 cup water
1/8 teaspoon salt
1/4 teaspoon cream of tartar or 1 tablespoon white corn syrup
1 1/2 cup sugar
2 egg whites
1 1/2 teaspoon pure vanilla extract

Place sugar, cream of tartar or corn syrup, salt, water and egg whites into a mixing bowl and beat for 1 minute with a handheld electric mixer. Using the double boiler method, place the bowl over the top of rapidly boiling water, making sure that boiling water does not touch the bottom of the top bowl. (If this happens, it could cause your frosting to become grainy.) Beat constantly on high speed with electric mixer for 7 minutes. Remove the bowl from the boiler and gently stir in vanilla with a spatula.

Recipe courtesy Paula Deen