We only have one scriptural record of Thaddeus, other than his name:
Matthew refers to him as "Lebbaeus, whose surname was Thaddeus." John calls him Judas, son of James. Likely, his given name was Judas, and Thaddeus/Lebbaeus were nicknames.
"Thaddeus" means "breast child." "Lebbaeus" means "heart child." Many have suggested the idea of a nursing baby here--maybe it was an insult--like a "mama's boy." The nicknames might suggest a tender, childlike heart.
John 14:21--"Judas (not iscariot) said to Him, 'Lord, how is it that you will manifest yourself to us, and not the world?'" On the surface, this question is gentle, and not brash or demanding. It was obvious that he still had not grasped the real reason Jesus was here. He still thought Christ's purpose was a political one--to reveal Himself as King.
Of course, we have no concrete evidence about Thaddeus' life after Jesus' ascencion. But here's the tradition:
A few years after Pentecost, he headed north and took the Gospel to Edessa, a region in present-day Turkey. There were accounts of how he healed the King of Edessa, a man named Abgar. In the fourth century, Eusebius the historian said the archives of Edessa contained full records of Thaddeus' mission and Abgar's healing. (unfortunately, those archives were destroyed). He is thought to have died in that region.
Sources: John MacArthur, "Twelve Ordinary Men"
Mercer Bible Dictionary
Class notes, "The Twelve" seminar, 1996
We did a series several months ago on the twevle; so all of this info was handy to give ye.