Step 1: Rules used
For an alcohol with a double bond, the principal chain must hold the OH, and the OH gets the lowest number. Next the double bond gets the lowest possible number, then substituents.
Step 2: Count the carbons
From the drawing, the main chain is two carbons (ethyl), one branching carbon, the two carbons of the double bond, the OH carbon, another branching carbon and a final ethyl group: $2+1+2+1+1+2 = 9$ carbons. This gives the nonene skeleton.
Step 3: Locants
From the right, OH sits at 4 while from the left it would sit at 6, so we number from the right. The double bond is then at C5, and the two methyl branches are at C3 and C7.
Step 4: Name
Writing the substituents in front, we get 3,7-dimethylnon-5-en-4-ol. A name with "ethyl" substituents and an 8 or 7 carbon chain (options A, B, D) ignores the longer 9-carbon chain.
Final Answer:
The name is 3,7-dimethylnon-5-en-4-ol. This is option (C).
\[ \boxed{\text{(C) }\text{3,7-Dimethylnon-5-en-4-ol}} \]