I found a family tree that shows a James Thomas Lynch Born: 1856 in Red River Co., TX 1885: Marriage to Amanda Elizabeth "Mandy" Suggs Died: late 1890s
If this is who you're looking for, you can contact the Red River County Clerk, www.co.red-river.tx.us/ and ask if they have birth certificates back to about 1856.
If not, you can contact www.dshs.state.tx.us/vs/reqproc/faqbirthdeath.shtm.
Have you looked for his and Amanda's marriage certificate? Maybe they married in Fannin Co. There is a volunteer in Fannon Co. who will go to the courthouse. You'll find his/her contact info at www.gengathering.com. You do have to create a user ID and password to enter this web site. However, access to the volunteer contact info is free.
Do you know where Tom died? P. S. You might want to consider adding your email address to your profile so volunteers can contact you directly.