First of all, D.H. Lawrence published this book under a few different titles, notably Tenderness and Lady Jane and John Thomas. What's the difference?
Tenderness is, as you might have figured out, a big deal for Lawrence. The whole point of Mellors's grudge against the modern world is that it has no tenderness. He begs for warm-heartedness, telling Connie that he believes
in being warmhearted. I believe especially in being warm-hearted in love, in fucking with a warm heart. I believe if men could fuck with warm hearts, and the women take it warm-heartedly, everything would come all right. It's all this cold-hearted fucking that is death and idiocy (14.182).
When Connie and Mellors come together, they treat each other with tenderness, not with their words so much as with their bodies. Calling the book Tenderness shifts the focus from the sex to the actual quality of relationship that Lawrence is saying the modern world needs.
In contrast, Lady Jane and John Thomas puts the emphasis directly on the sex. "Lady Jane" and "John Thomas" are the couples' nicknames for their genitalia, and what happens with these names is that they become nicknames for the whole person. In other words, Mellors becomes totally identified with his penis, and Connie becomes identified with her vulva. This substitution of a part for a whole is a literary technique with a nice big word: synecdoche. But, since "tenderness" seems to be indistinguishable from sex for Lawrence, there isn't necessarily much difference in terms of what he's getting at.
So finally we come to Lady Chatterley's Lover. Well, notice that it's not called Lady Chatterley, even though she's the primary focus of the book. That's a little weird. What to make of it? First, D. H. Lawrence is a bit of a misogynist. We know that already. He's interested in Lady Chatterley, but really only in her relationship to other men. In fact she says at one point that she's just relieved to be free of women: "Ah! that in itself was a relief, like being given another life: to be free of the strange dominion and obsession of other women. How awful they were, women!" (17.10).
Second, there are no names. Lady Chatterley isn't a name; it's a title. And "lover" certainly isn't a name. "Connie" and "Oliver" as individuals aren't important. That's a point that Lawrence makes multiple times. Actually, Mellors's first name is only mentioned once or twice at all, and Connie has a hard time with him at first because she thinks he's not "quite individual enough; he might be the same with any woman as he had been with her. It really wasn't personal. She was only really a female to him" (10.125).
Third, and finally, there's a friendlier way of reading the title. Because there's a possessive—Lady Chatterley's Lover—the two are maybe even more connected than they would be if Lawrence had stuck with the "and" of "Lady Jane and John Thomas." With an "and," there's no necessary connection. It's just two people stuck together. But "Lady Chatterley's Lover" shows two people joined by possession. (Of course, there's also an unpleasant little class thing going on, almost as if he's still her servant.)