I still suspect the accuracy of your reading. That said, a HR of 30 is a matter for concern, at least enough to get it checked. The problem being, if it happens only when asleep (maybe a particularly restful sleep) the doctor will not be able to measure in the office. A Holter (24 hour monitor) which uses, I believe paste on pickup, just like an EKG, could be prescribed, but it is an expense you may not want to incur until you are sure your reading is accurate. Maybe just testing again for a few nights and see what numbers you get.
If you feel fine, no dizziness during awake times the low sleep HR may not be a problem.
I have a brand new Timex HRM with a chest strap and it does have storage capability. I only sleep on my back because I have had back issues in the past and cant sleep on my side or stomach.
A resting HR of 30 is in the range of "too low"..but I'd question the accuracy of the measurement. If you were in fact asleep, you may have been in a position that interrupted your heart rate monitor. I am not familiar with what you have, my exercise heat monitor uses a chest strap to detect the pulse, I know if I have that on while laying down I can get in a position on my side that the chest strap isn't making good contact.
I assume your monitor has a storage capability, storing low and high reading, for example. Mine just reads real-time.