I forget the name of the High Rock Septims, and along with that, what information is known about them?

Apart from them, who else in High Rock is in a good position to claime the throne since the Emperor's death?

Probably the disenfranchised branch that some think may have killed Katariah I, if they still exist.

I think the OP is talking about King Lysanders decendants, I seem to recall reading in some Daggerfal literature that he was a distant cousin of the Septim line. I would imagine that most of the high ranking noble families have some Septim blood, or at least family lore saying they do. But without the Amulet of Kings, any Tom, dike or Harriet could claim they are a Septim and there is no way to disprove it.

"Not even death can hide a true descendant of Tiber Septim from the Underking."

That line is spoken by one of the Underking's "agents" during the storyline of Daggerfall, I believe during the quest where you set out to acquire an artifact from Castle Llugwych for them.

I wouldn't know about the quest, I was reading thru the storyline on The Imperial Libary site, so yeah King Lysander was a Septim so his son would have a legitimate claim on the the throne, if his son is still alive, not having played the bleeping game I wouldn't know. I would give anything to get my hands on a working copy of that game!

The Kings of Shornhelm are Septims by law, but not by blood, being descendants of Andorak, the son of Emperor Uriel (I dunno which number, too lazy to look up), who in turn was son of Empress Katariah (a non-Septim) and had been adopted into the Septim family.

Its the family Ive regularily advocated in post-OB suiccession matters.

His son, Gothryd, is the King of Daggerfall during, well, Daggerfall. I think he's still alive by Oblivion, due to this excerpt from the 3rd Edition of the Pocket Guide to the Empire:

"King Gothryd and Queen Aubk-i's own marriage twenty years before had cemented relations and formed the basis of the peace between Daggerfall and Sentinel, which continues to this day."

Of course, it could just be referring to the peace relations that "continue to this day", but I doubt they're dead by now.