While the dose you gave (12.5 mg diphenhydramine) is above any indicated dose for a 15 pounds, 8 month infant, the risk of severe adverse reactions is low. Having said that, it is important to be aware of and look for any of the following side effects:
If you notice any of these, you should bring the infant to the emergency room.
As mentioned above however, the risk of severe adverse effects are low. While dosing Benadryl isn't actually recommended in under person under the age of 6 years old, the dose of 1 mg/kg is commonly used in studies for infants with sleeping disturbances. For a 15 pounds infant, that comes out to a dose of 6.8 mg.
Similarly, studies have used the dosage of 2 mg/kg/dose for the treatment of anaphylaxis in infants. For a 15 pound infant, that comes to a dose of 13.6 mg. It is important to note that Benadryl is not considered a first line therapy for anaphylaxis and should not be used as the sole agent for treatment.
Lastly, the maximum daily dose that has been safely studied in infants in 5 mg/kg/24 hours. This equates to 34 mg for a 15 pounds infant.
So, while the dose given in this case (12.5 mg) is over the recommended dose of 1 mg/kg (6.8 mg), it is under the dose that has been used safely for the treatment of anaphylaxis (2 mg/kg or 13.6 mg in this case).
Based on the information available, the dose given should not cause serious harm. It is important to remember that the FDA, and the maker of Benadryl, does not recommend use in children under 6 years old unless directed by a doctor.