What an exciting discovery! :D The Tyner surname indeed comes from an Anglicized version of the Gaelic name "O Teimhneain," which is derived from the word "teimhean," meaning "dark." The name was first found in County Laois.
Have you been able to find any ancestors with a link to the Emerald Isle yet? You can find a helpful thread here
if you want to start with your research. :)
You are also more than welcome to contact an organization called Ireland Reaching Out
, where great volunteers attempt to connect Irish descendants to their roots and family members, through reverse genealogical research.
Feel free to post any questions here about your research, as I'm sure our members would love to help out!