Is there a way to add a note to the iOS Health app?

Adding notes in Health.app

No, as of iOS 10.3.x you can't add comments to Health.app and its database.

Alternative (3rd party apps)

If you want to enter any notes or observations you need to rely on a third party app to do so.

Ideally, this app should be able to export your notes in a non-proprietary format like XML. Not all apps (e.g. Pillow) with a notes feature allow you to do that. (Thanks @Mark)

Alternative (use unused fields)

You may use unused Health.app fields to store your observations. Below are special data types that exceed the usual 'one value + one time stamp' values.

With graphs

• UV Index (one number, start time, end time)
• Sleep Analysis (three values, start time, end time)
• Blood Pressure (two numbers)

Without graphs

• Carvical Mucus Quality (five values)
• Menstruation (four values + one boolean (yes/no))
• Ovulation Test Result (three values)
• Sexual Activity (three values)

You could totally add 'UV Index' (value + duration) for headaches and 'Vitamin B12' for forgotten medications. Add them to Health.app's Today view using the 'Add to favorites' toggle and you're done.