How much do you know about your HVAC system (type of refrigerant used, how the duct work splits off the supply air plenum and runs to the room with the odor)?
A short answer is that strange, chemical-like odors can emanate from HVAC systems from both the refrigerant and the lubricating oil that circulates throughout the system (the compressor that is outside in the condenser unit needs continuous lubrication which actually circulates through the system with the refrigerant). It's also possible to have a very small / slow leak that doesn't materially impact system performance in the short term, but still produces the odor (more on that below).
The long answer is that most residential HVAC's use chemicals for refrigerants that do have some kind of chemical odor (hydrofluorocarbons smell like SOMETHING to most people, although smell can be quite subjective). Also, some of the oils have odors as well. If you have an evaporator leak, it's not impossible that a small amount of refrigerant is leaking out into the HVAC air handler while the unit is down and then gets "flushed" into the room when the unit turns on. Refrigerant compounds are gases, but they are heavier than air meaning they would preferentially pool at the bottom of the HVAC air handler and then get flushed preferentially into certain parts of the home.
For example, I know from experience that the refrigerant in my vehicle has an ether-like odor that reminds me very much of fresh cut grass. So if I smell fresh cut grass in a large, parking asphalt/concrete parking lot, I suspect an evaporator core leak. I also know that my vehicle uses poly-alkylene glycol oil (PAG oil) for compressor lubricant which to me smells like latex band-aids, again from experience.
If you have a very slow evaporator leak, those chemicals may leak out slowly enough so the system still has enough refrigerant charge to cool your house right now. That could change say in a year as the leak continues or if the leak starts to grow. A very small leak could allow refrigerant to accumulates in the relatively closed HVAC air handler / supply air plenum over time while the unit is down and then get pushed out all at once if the unit cycles on which might produce enough locally for you to smell. Since the refrigerant is heavier than air, it's not impossible for it to get selectively pushed out into one area of your house since your supply air plenum probably looks like an octopus with supply ducting coming out in different locations and at different heights on the plenum.
Do you detect the odor every time your HVAC turns on? Or only at certain times, like after the HVAC has been off for an extended period of time? To me, the smell showing up only after extended periods off would support the really slow leak possibility as it would allow for more accumulation / higher concentration pushed out when the unit does cycle on which you are more likely to detect. If you smell it every time, I would look elsewhere as that would be a large leak and very quickly you will run out of refrigerant charge for the system to work. The easy look out here is HVAC run time cycles increasing / increasing cold supply air temps.
If you do think it's possibly refrigerant, your HVAC guy should be able to put UV dye in the system to pinpoint the leak with a blacklight. If you are more DIY inclined, there are relatively cheap, handheld electronic refrigerant detectors these days that pick up fluorocarbons that should go off if it is refrigerant and it's strong enough that you can smell it. Both of these options would be relatively cheap checks vs. jumping off on a HVAC system component replacement.
Full disclosure, I say all of this having never had to personally deal with a residential HVAC system leak. I have dealt with multiple HVAC refrigerant leaks in vehicles and also have spent 15 years working in chemical plants troubleshooting those systems and solving problems not completely dis-similar from this. Good luck.
