It is not unusual to wake up with an achy or sore feeling on just one side or in one particular area. What your daily activities are, and how fatigued you are may affect how you feel or sleep at night. Do you sleep in any one particular position? If not, do you tend to roll around in bed a lot, until you find a restful position? Since you relate this to waking up, it is most likely a positional situation with you laying on that side more often.
I would advise you to see a primary care physician
so that you can go over your specific symptoms in greater detail. Your doctor
will be able to ask specific questions and tailor a diagnosis based on those symptoms. A physical examination of your musculoskeletal system is also essential in providing a clue to what specifically is going on. If you are working out, you may want to think about whether you focus on your right side more so than your left or if you are lifting heavier weights etc. Generally such aches are related to muscle fatigue and overwork and will usually self-resolve themselves.