OK, I realize this may be an unpopular idea. But, hear me out.
O'Connor is a moderate. She stated in the Casey v Planned Parenthood that the Court should not overrule Roe because it was the law for 20+ years and was now part of the national fabric. In addition, she is consistantly the one justice who tries to forge a moderate course between the far-right and the far-left of the Justices. Yes, I know she voted for Bush in the 2000 case, but the bottom line is she is the only person in the majority who is palpable to moderate Republicans. The Democrats will need their support to push this through.
No, she is not a crazy wingnut. Much of criticism using this argument is from people who have not been trained to read SC decisions, and is instead based solely on her party affiliations.
It would be a way to appeal to moderate Republicans. The far-right is screaming at fever pitch right now, bullying the moderates. The Dems can quietly play this as a way to lobby the moderates to either join with the Democrats on more issues or switch parties (unlikely, but a possibility).
She is a woman. The Democrats need to start refocusing their stance on women's rights. Kerry took single women by 63-36. We need to improve this margin by at least 10 points. The Dems could use the idea of women's rights as a smoke-screen for abortion/reproductive rights.
She is the only judge with Legislative experience. She was a state legislator before she was a judge. She understands how to lobby people and brign people together. No one else on the court brings this experience to the court.
The Dems can cast Scalia or Thomas as a Radical. By stating we support moderates, we are implying the Republican choice is a radical.