While speculation about what Mab would become without her mantle is interesting, I want to get back to the OP. However, for the record I will state that Mab only has a trace of her original humanity left in her, so it's difficult for me to see how she would survive let alone function without her mantle, beyond taking on another mantle, like Mother Winter's or that of another immortal.
Now to address the OP. Near the beginning of Battle Ground, Mab tells Harry that the first time she saw him, Mab could tell that Harry had potential for true greatness. By that, I think that Mab saw Harry's potential to become an immortal. When we look back to Summer Knight, we find that Mab told Harry that she had seen him frustrate the Leanansidhe the previous autumn. Mab did not say that was the first time she saw Harry, but it's good candidate for first time that she saw him. At that time, Mab saw Harry outwit a particularly clever and powerful immortal. Mab may have thought something like, "Imagine what this one could do if he lives long enough to gain knowledge, wisdom and gets the power to really use it." Of course, that doesn't get us any closer to why Mab wants to change Harry; except it tells us that Mab was looking for someone who is clever enough to match wits with other immortals; perhaps she wants someone who is something of a trickster or in modern human terms, someone who can think outside the box.
Two questions come to my mind at this point. First, who does Mab want Harry to outwit and how does Harry become an immortal and take or create a mantle for himself. The second question is easier to answer, so I'll start with that one.
We saw that Lilly had the mantle of the Summer Knight placed in her by Aurora. Then, when Aurora died her mantle jumped over to the nearest reflection of Summer, which was Lily. So by extension, Harry is already prepared to take up and put on the mantle of a member of Winter Court. I'm assuming the mantles of the Winter Queens are off limits to Harry. So, without other possible candidates the most likely mantle Harry could inherit would be Kringle's. Add to that, Kringle told Harry that "Many, many mantles are worn - or discarded - on Halloween night, wizard." Potentially, that makes it much easier for Harry to become an immortal. All he needs is for Kringle; or another member of Winter that has a mantle that would be available to Harry, to willingly discard it and for Harry to be the closest reflection of Winter. Yes, Harry could kill another immortal, but we know how difficult that can be, and Harry would have to do it on his own to make certain the mantle passed to him or no one else could pick it up; and why would Harry want do this in this first place? It's much easier for Mab to force the choice on Harry to pickup a discarded mantle than to force him to choose to kill an immortal and take their mantle.
Of course, there could be rules around how mantles are passed from one party to another that we haven't been told yet. Perhaps Kringle; and therefore Odin, has to die for his mantle to automatically fly to the closest reflection of Winter. If Odin were to discard Kringle's mantle, perhaps it would take an effort of will to pick it up so Harry would have to make the choice to do so. We just don't know. I assume that for any discarded mantle that comes from outside of Winter, Harry would have to choose to pick it up. Of course Mab and Odin are masters at creating conditions that would pretty much force Harry; or any other mortal, to make such a choice.
My guess is creating a new mantle is something that usually takes ages. For example, I doubt the Red Cap has a mantle. Perhaps if he and his legend survives and grows over the coming centuries, then maybe a mantle will grow around him. If this is the case, there's little point thinking that Harry might create a new mantle. I suspect that Harry would have to develop almost god-like worship from a huge number of mortals to start building a mantle for himself, and why would he even want to do that. I don't thing Harry proclaiming himself as "The Wizard of Chicago" is going to do it. Harry could also try to pull-off a darkhallow spell, but that's not something Harry would want to do or could be convinced to do.
So, my best guess is Harry will get the opportunity; whether he likes it or not, to pick up a discarded mantle, and it will probably be from Winter, because this way Mab can keep Harry "in the fold" and at least partially under her control. A sidetrack issue is created no matter where the mantle comes from. Can Harry remain the Winter Knight; the mortal champion of Winter, if he picks up a mantle from an immortal? I think he can and it should be pretty obvious why. Harry could make the choice to live in the mortal world; which makes him nominally mortal in that he could be killed, just as Vadderung can be killed in the mortal world. If Mab sends Harry on a mission he couldn't use his immortal abilities, because Harry would no longer be the mortal champion of Winter. I don't think Mab would like this, because every time she sent Harry out on a mission she would be risking both Harry and the immortal mantel he carries. Then again, there might be rules we don't know about that lesson this risk; I guess we will just have to wait and see.
Finally back to the "why" question, which involves who Harry would take on. The obvious candidate for immortal Harry to have to outwit is Nemesis itself. Harry might take on the Walkers individually as a mortal, but for the big boss Outsider I suspect Harry needs to level up in a big way. After Nemesis there is the Prince of Darkness, though I don't know if we will be seeing him "in the flesh" so maybe not such a good candidate for sometimes-immortal Harry to face. Plus, if the mortal Daniel Webster could out argue Satan, I doubt Harry needs to beat Satan to a pulp or anything like it. Then there's Drakul. We don't know how powerful he really is. While Drakul is on earth he might be vulnerable, when he's in the Nevernever perhaps it would take god like powers to tackle him. Facing those possible opponents are my best guesses why Mab wants Harry to level up in a big way.
I think the only other reason that might be motivating Mab, is she senses that her time may be coming to an end. I'm reminded of a phrase from the movie The Usual Suspects, "It was his Will!" Mab wants her "Will" to continue beyond her time, and to that end Harry may be the best instrument available to make that happen, but only if he becomes an immortal. Even if Mab knows that she won't be there to see it, in Mab's mind she and Harry would be accomplishing "great things" together.
