Assuming you mean the MIT-X11 licence, this requires that:
The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
So yes, you do need to preserve their copyright notice, and the text of their licence, even in the shipping binary. This is generally pretty easy in Android apps, which are by their very nature graphical and amenable to drop-down, nested menus.
You ask for extant examples. Poking through my phone (which runs only free software), I find that for a small app, a good example is Simple File Manager Pro. Under
Settings (...) -> About is a menu entry
Third-party licences which has all the required copyright and licence statements for the packages on which it depends, and has links to them in addition.
For a larger app, Wikipedia has under
Main Screen -> Menu (top left) -> About a section on Libraries Used, with a clickable in-app license statement for each one as well as a link to that project.
There's nothing special in those choices; they're just examples of free software I already have installed where I think the authors have done a good job honouring the exact terms of the (non-copyleft, free) licences on elements they've used to make up their code. I should add that I also managed to find a fair number of installed free apps that don't realise that more than a link to each contributing project is needed. Thee cheers for you for not wanting to be such another one!