Ok...first I laughed this off and took it more or less as a joke, but now it's really intriguing me. xd
LastMistBloodline (you are the one that pumped me up*laughs*), Hyuugas supposedly being the ancestor of the Uchihas has nothing to do with being dominant or not, but what's important here is the actual individuals, in this case Hinata and Sasuke, and which gene is stronger than the others. The whole lineage doesn't count(at least not 100% ).
You said that Uchihas interbreed in order to strengthen the gene, while Hyuugas taking outsiders will have quite the opposite effect(just like in the mangas when breeding a demon and a human, results a semi-demon; kinda like that)
I intended to base my answer on
actual Genetics(I luved it in high school :3 ) , but this was rather futile, judging by the fact that red and white doesn't have an equivalent in this area (I wonder why

), thus if I would've known which one is dominant and which is recessive, then it would have been easy to determine what type of eyes the kid will have.(for example, the brat will have Hinata's blood type.What's a nerd to do when he doesn't have what to do?? Stupid stuff,hence this! XD )
But seeing as black eyes, for example, are dominant over blue eyes (if NaruHina will happen, the kid will have her eyes color!Sad >.<), then my guess is that red (Sharingan) might be dominant over white (Byakugan). At least, from the viewpoint of the phenotype, this might be possible.
Btw... XD Did it crossed your mind, LastMistBloodline, that the eyes might look like Sharingans and have the Byakugans skill, or viceversa? xd
This seem twisted, but not impossible *laughs*