Regina and Naomi were lovers, though this was never explicitly stated.
This is ABC. It's Disney. We will never get confirmation.
Some things about this show are just not going to be cut and dry.
Too much circumstantial evidence points to it.
Regina was never sick. She killed herself out of grief over Naomi's death.
She never had a nosebleed.
She never showed signs of temporal consciousness travel.
No, the Cabin Fever and the consciousness travel were two different events. As much as I want to believe this theory, because it would be nice to have lesbians on the show, the evidence from Captain Gault suggests Regina was not an isolated case, and he mentions that he thinks Keamy might be suffering from the same effects. As someone else pointed out, none of the Cabin Fever sufferers appear to have jumped through time or indeed suffered from nosebleeds or brain aneurysms. Regina didn't die, she killed herself after her mental state deteriorated, which would be evidence for a partnership between Naomi and her had it been a single case.
In some circles (military for example) 'sister' is a euphemism for lesbians, if in an environment where one does not wish to be public about their choice.
Also compelling, but the problem with this is "Tell my sister I love her" has already been explained by Miles to be code for "I'm being held at gunpoint." It's possible that either Miles made this up, Minkowski understood that Naomi was not using code and was referring to Regina, or it simply presents an irresolvable problem for this theory, in addition to the flaws with the "Regina became depressed over Naomi's death" theory.
We were never told her last name; it may have started with a G.
Regina didn't seem to have any similarities to what happened to Rousseau's team, and she hadn't even been on the Island.
Regina was trying to get back to the "other side".
The book she was reading was upside-down, suggesting that she was mentally stuck in the "upside down" realm (starboard-realm, dark-side, if preferred).
Her book was upside down because she was experiencing a mental breakdown as a result of the Cabin Fever, or - if the theories in the above section hold any merit - because she was in love with Naomi and mourning her loss.
When she "committed suicide," she was actually trying to get back to the other side. As we know, the most effective way to get to the Island was via submarine, under water.
By wrapping herself in chains, she was ensuring that she would sink fast enough to make it through to the "other side."
By wrapping herself in chains she was trying to weigh herself down enough so she could reach the Looking Glass station (the same way Charlie did).