Something tells me mRNA is irrelevant. It's a common talking point among anti-vaxxers, and is typically nothing more than an excuse. It's also a form of gish-galloping, where they pile a bunch of bullshit on you and make you defend it.
Ask them some follow-up questions like these. I suspect the trend will become clear.
What are your thoughts on the more traditional non-mRNA covid vaccines, such as the ones from J&J or Novavax (or whatever you have in your area)?
When was the last time you got any vaccine, including a flu shot?
If you had the choice today, would you get the well-established vaccines such as polio or measles, mumps, rubella (MMR)?
Once you have these answers, you'll know if they are truly concerned about mRNA being new or if it's something else.
Yes. Pfizer and Moderna made mRNA vaccines. Johnson & Johnson and Oxford-AstraZeneca made DNA vaccines with an adenovirus vector. Novavax made a protein subunit vaccine.