Irrespective of one's religious beliefs, a genome from the mid-1st century AD would be interesting. I am personally in favor of the scientific study of saints' relics.

Bulgarian bones could be John the Baptist's, scientists say
So when Bulgarian archeologists announced in 2010 that they had found the bones of John the Baptist, Tom Higham was skeptical.

He got a surprise.

Higham, an Oxford University scientist and an atheist who doesn't believe in "any kind of religion or God or anything like that," was asked to test six small bone fragments found on an island named Sveti Ivan - St. John.

The bones turned out to be from a man who lived in the Middle East at the same time as Jesus, Higham said.

"We got a date that was exactly where it should be, right in the middle of the first century," said Higham, a radiocarbon dating expert.
...

DNA testing by colleagues at the University of Copenhagen suggested that the person was most likely to have been from the Middle East, he said.
"We have a complete genome. It's possible that we could step this a step further and see if there is any similarity," in the genetic material of all the relics.
St. John's father Zechariah belonged to to the Aaronic line of priests. While modern Jewish priests (Cohanim) belong to multiple Y-chromosomal lineages, I think it's a good bet that at least one or a few of these lineages could be traced to Jewish priests of 2,000 years ago (leaving the question of the ultimate Aaronic lineage open). If the remains from Sveti Ivan island match one of these lineages, then this would be a powerful piece of evidence in favor of their attribution to St. John the Baptist.