Which of the following is a common side effect of thyroid replacement drugs?

Tachycardia is a common side effect of thyroid replacement drugs because these medications work by increasing the level of thyroid hormones in the body, which can stimulate the metabolism and the heart rate. When patients are prescribed levothyroxine or other thyroid hormone preparations, it's essential to monitor their heart rate, as an excess of thyroid hormone can lead to an increased heart rate or palpitations.

Other options such as weight gain and low blood pressure are typically not associated with thyroid replacement therapy; in fact, many patients actually experience weight loss as their metabolic rate increases. Dry skin is also not directly linked as a common side effect of thyroid replacement drugs; rather, dry skin can be a symptom of hypothyroidism itself, which is what these medications are treating. Therefore, tachycardia stands out as the most relevant side effect in the context of thyroid replacement therapy.