Do the Starks have claim over those two swords? The were made from Ned's sword. 
Nope. They got their shit jacked. It is what it is and no one would recognize their claim anyhow. I doubt it is even common knowledge outside the Lannister circle that Ice was melted down to forge them within the canon.
Also, someone refresh me... does Lady Stoneheart and the BWB recognize it solely as a Lannister sword, and not realize it came from Ice (i think this is correct) or do they know it was forged from Ice? Is they knew that, I feel like Lady Stoneheart would have had a rage blackout and not even given Brienne a choice.
kennyfrommd -Also, someone refresh me... does Lady Stoneheart and the BWB recognize it solely as a Lannister sword, and not realize it came from Ice (i think this is correct) or do they know it was forged from Ice? Is they knew that, I feel like Lady Stoneheart would have had a rage blackout and not even given Brienne a choice.
No, I don't think they knew it was forged from Ice.
I think they see the Lion's head on the pommel and assume she is an agent of the Lannisters.
And that is a death sentence from LSH