I would definitely choose the Byukugan.
- the Byukugan can see miles ahead.... need that ability to avoid unneccessary troubles
- to see through objects and obstacles to check if there is anyone hiding around or if I wanna find where my opponent is hiding
- I can tell if that person has done a replacement or clone technique because of the difference in chakra and stuff. This means I don't have to waste my shurikens or other ninja tools on the decoys
- I always have a thing for absolute defense stuff...
- Ability to control chakra flow.... how cool is that? I can even use it to increase the flow on myself
- The speed that comes with being a Hyuuga (hmm I'm not too sure about this but it does seem so).
About the Sharingan's genjutsus... don't think there's much the Byukugan can do against that. I just figured that having Sharingan makes one a Jack of all trades but a Master of none.... unless ur a super genius like Neji is, to learn Hyuuga's secret techniques on his own. At least Sasuke had tutors like Kakashi and Oro to help him advance his abilities.