Our Icon is an "H" inside an SE bubble, this is identical to the icon of the already existing History.SE.
The Icon for the Health site should be changed to avoid confusion.
Nice catch, didn't even notice. Obviously we will have to change our logo because we are the newer site. I have a few ideas for a new logo. We could just do "HE", but that is a little boring. I was also thinking we could possibly do a red cross, but that would conflict with Christianity a little. I think that a heart might be good. I don't think this would conflict with any other sites and it would probably look good too.
What I now recommend, is to do a heartbeat icon. Some people might confuse the heart for romance, but the heartbeat icon is obviously related to health. (Idea from Zach Saucier in the comments)
I've decided to try to design the heartbeat logo, so here it is. Let me know what you guys think.
More accurate heartbeat made by @Tim, red and white versions (the small size gets cutoff)
If you would like to improve it, here is an SVG link: https://openclipart.org/download/216852/heartbeato.svg
I would suggest a icon version of a nice ECG of a heart in normal sinus rhythm.
Good sinus rhythm being usually seen a good health symptom.
Source of image "SinusRhythmLabels" by Created by Anthony Atkielski. Licensed under Public Domain via Wikimedia Commons
How about one of the hospital sign.
The H sign could work if it's not as light-blue as the history, and that it's lacking the talk balloon, and having it look more like a hospital sign. Plus it's obviously referring to Health as well.
Anyway personally I like the plus sign more.
Maybe just make a SE icon template (with balloon that wraps the plus), to make it SE-specific. I think the dark-blue color emphasizes it's something else.
And I'm sharing you here with Tim's suggestion as well (ignore the black background):
Again the idea could be nice but the plus sign will barely be recognizable as a favicon. Perhaps changing increasing the plus sign could work.
IMHO the final icon should have one main idea leveraged by all the the awesome ideas suggested in this thread.
I propose that the icon should use the Rod of Asclepius which is the serpent-entwined rod in the middle of the Star of Life: