Sivir is part of Azir's bloodline?

Something like 5% of Mongolians are direct descendants of Genghis Khan. Who knows, maybe Azir had a particularly large harem.

It would also help explain why her weapon was the key to the tomb and activated the booby traps and everything.

Technically it wasnt even directly stated as to how close their relation is. I mean if you take a guy from ancient times, he definetely has a shitload of people related to him after hundreds of years.

What we DO know for sure that Azir saw in Sivir himself, so it might be that its not the blood relation that is the most important, but the spirit of one...

Odds are someone is his descendant, youngest of the bloodline and therefore the last (last as in time, not amount). Then odds are, in a fantasy world that person is attracted to the ruins of Shurima because of their bloodline. Then they get known for exploring the ruins, hired and subsequently betrayed by Cassiopeia, taken back from the brink of death by Azir and then famous for being related to the ascended ruler of Shurima. Considering Runeterra is a dangerous place they picked up some combat skills, enhance this by being heir of an ascendant (probably brings some advantages, look how many champions get their powers from magical items).

This champion happens to be Sivir.