Not to mention impeachment. To become impeachment-proof Hillary's best bet would be somebody from the left, like Warren or Bernie.
I've never really thought the "impeachment insurance" factor in selecting a running mate was a big deal, just in general terms. And in the specific context of 2016, I can't see it as much of an issue. The House will very likely remain under Republican control and there wouldn't be anything to stop them to bringing impeachment charges, but a conviction in the Senate requires a two-thirds vote. Even if the Democrats don't regain control of the Senate, they're going to have a minimum of 46-48 seats in the Senate, so a conviction would require a minimum of about a third of them to defect, and that isn't going to happen without wrongdoing so egregious that no amount of "impeachment insurance" could save the day.