**Core Concept**
Failure to thrive and delayed teething in a 1-year-old child can be indicative of several systemic and genetic disorders. The condition highlighted in this scenario is known as **Hypodontia**, which is a rare dental anomaly characterized by the absence or underdevelopment of teeth.

**Why the Correct Answer is Right**
Delayed teething in a child can be due to various reasons including genetic factors, systemic diseases, or environmental influences. In the provided options, we will identify the correct underlying cause of delayed teething in a child. Option C is correct because **Congenital Syphilis** is indeed a possible cause of delayed teething in a child. It is a well-known congenital infection that can affect multiple systems, including the dental system, leading to delayed teething.

**Why Each Wrong Option is Incorrect**
**Option A:** **Hypothyroidism** can cause delayed teething in a child due to impaired thyroid hormone production, which affects the development and growth of teeth.

**Option B:** **Cleft Lip and Palate** can also cause delayed teething in a child due to the anatomical abnormalities that can affect the development and growth of teeth.

**Option D:** **Down Syndrome** can cause delayed teething in a child due to the genetic abnormalities associated with the condition, which can affect multiple systems, including the dental system.

**Clinical Pearl / High-Yield Fact**
When assessing a child with delayed teething, it's essential to consider the possibility of systemic diseases, genetic disorders, or congenital infections that can affect dental development.

**Correct Answer:** D. Down Syndrome.