What I have learnt from my own family tree is that Lyman was Leman in the early 1000s CE and before that it was Leofman. We originally came as Saxons from the Netherlands to England at around 800 CE after the Romans left. I have traced my tree back to 1000 CE and the first few (oldest) were Leman.
By the way, Leofman means 'lover man', which derives directly from Old Saxon and Dutch and those came from Proto-Germanic language originally; Wiktionary has a good description of Leof.
I have read from other sources that there are very many variations of Lyman, including perhaps Lehmann, which is also of Germanic origin.
My genetic results from many different companies all point to my being German and most like the modern day Dutch people, yet my Lyman line came directly from England in the 1600s to the US. In fact, the first registered death and one of the founding members of Hartford, Connecticut was my Lyman ancestor. This all fits perfectly with my surname origins. Hope this helps you and/or others : )