We already got a vague idea of what Big Mom power can be and some people already trying to name it with Sei-sei no Mi or anything related with life.
Now, I want to give my input on his power.
Her DF power name is Tama Tama no Mi, Tama from the word Tamashi which translated as Soul and can also be translated as ball.

Now we go to the 'why?' part.
1. Well, she does looks like a ball after all with how round she is.
2. Her power related heavily with soul.
3. The first thing we knew about her is, She didn't have a ball! Coincidence?

Now this is my main point, her combat ability.
4. Some of people already stated on the internet while her DF power is really dangerous and stuff, it's not something that can be useful in battle or something that can enhance your fighting abilities, well, comes to think of it, they did have a point and that is why I chose the word Tama Tama as her DF name.
Because she can use Kotodama https://en.wikipedia.org/wiki/Kotodama
Well, to put it simply, Kotodama or Kototama mean the soul of word, it's an ability which makes you capable to give an absolute 'order' to any living being with a single word, e.g Bind, Scatter, Blast, Sit, Stop and etc, people who watch Noragami or anime about Onmyoji maybe familiar with this term.

But what's the basis of this theory?
Because Big Mom really emphasis the 'my order is absolute' kind of vibe, well she is a Yonkou it's not something that would surprise anyone but one thing I realize is the fact that she take it to another level.
In Kaido case, when Joker failed to fullfil Kaido wish to make an all DF user crew, Kaido just let it go and thought nothing about it and just said "you're just too damn weak Joker."
In Mama case.
-Her sweets tantrum, no matter what happen she need to eat the thing she want to eat and anyone who didn't 'obey' it will be killed just like our poor Muscat, may god bless your soul Muscat.
-His Tea Party invitation is absolute.
This is what intrigue me the most, Pekoms said "When someone invited to Mama tea party even God himself must attend"
Is this just a figure of speech or Big Mom really had the power to even force God to obey her order because she can use Kotodama?

If my prediction is true then she would be one hell of an overpowered opponent because, according to Japanese mythologies, there is a soul RESIDE on everything, and if that is the case then her power will not only limited to a living being and can literally order everything around her.
Why did I come to this conclusion is this page, on the first panel, you could see something...blazing? fire like thing come out of the ground in front of Big Mom, this is just hypothesis but maybe this is the soul of the thing she destroy and hence why I'm convinced that Oda following the Japanese mythologies of there's a soul RESIDES on everything in this world, and Mama is capable of using her power on it.