**Core Concept**
The clinical presentation of a breast mass in a young child, particularly in the context of a family history of breast cancer, requires a careful evaluation to distinguish between benign and malignant causes. This scenario highlights the importance of differentiating between juvenile breast masses, which are often benign, and those that may indicate underlying malignancy or genetic predisposition.

**Why the Correct Answer is Right**
The presence of a breast mass in a young child, especially with a family history of breast cancer, warrants further investigation to rule out a possible genetic predisposition, such as BRCA1 or BRCA2 mutations. However, the majority of breast masses in children are benign and can be managed conservatively. The clinical description provided suggests a possible case of juvenile fibroadenoma, a common benign breast tumor in children and adolescents. This diagnosis is supported by the mass being firm, smoothly circumscribed, and slightly eccentric under the areola.

**Why Each Wrong Option is Incorrect**
**Option A:** This option is incorrect because it implies a more aggressive approach, including imaging studies and biopsy, which may not be necessary for a juvenile breast mass. While imaging studies and biopsies may be useful in some cases, they should be reserved for masses that do not have a characteristic benign appearance or those with suspicious features.

**Option B:** This option is incorrect because it implies a genetic predisposition without considering other possible causes of breast masses in children. While a family history of breast cancer is a concern, it is essential to evaluate the child's individual risk factors and clinical presentation before recommending genetic testing.

**Option C:** This option is incorrect because it implies a surgical approach, which may not be necessary for a benign breast mass. In many cases, juvenile breast masses can be managed conservatively, with regular follow-up and imaging studies as needed.

**Option D:** This option is incorrect because it implies a diagnosis of breast cancer without considering other possible causes of breast masses in children. While a family history of breast cancer is a concern, breast cancer is extremely rare in children and adolescents.

**Clinical Pearl / High-Yield Fact**
In children and adolescents, breast masses are often benign and can be managed conservatively. However, a family history of breast cancer should prompt further investigation to rule out a possible genetic predisposition.

**Correct Answer:** C.