Not just Dragon Lance.
I can not quote the source, but I do remember reading somewhere that Tiamat kept dragonborn and half dragon servants in her divine domain at the gateway between the first and second layers of the Nine Hells.
The PC could have traveled to Dis on an errand for the servant order and ended up planeshifted.
Of course, stories of gods and goddesses having dalliances with mortals are common in myth.
I am all for cool backstories. My only concern would be a player trying to trojan horse in a demigod. Of course, Tiamat is not known for being a very caring goddess and probably has little interest in her offspring. And, while the dragonborn may have potential for greatness, demigodhood is 20 levels and a few epic boons way down the road. Even Hercules, son of Zeus, had to start somewhere. That is, unless you are running epic level. Then, by all means, give the dragonborn a temple he or she can see and hear in their mind, max hit die and a cell phone with mommy's number. Mommy's busy, though on a few hundred prime worlds and has thousands of unappreciative winy brats over the eons, so even as a demigod, dragonborn offspring is on their own. She may even spank the offspring for being weak and calling her too much.
Have fun!