Sounds fine to me. Especially if "their name is Lynn" or "Their name is Alex" or "their name is Leslie". If the name is gender-ambiguous, it makes sense to go with a gender-neutral pronoun.
If the name is strongly gendered ("Brett", "Suzanna", etc.) then it's a little weird because the information is already (presumably) there.
But Miriam-Webster's already ahead of you there (three years, IIRC). Check out definition 3: https://www.merriam-webster.com/dictionary/they