Although the Gospel Library app doesn't have a random scripture lookup button, I created a Random Scripture Generator website that does this exact thing, and it works well on both the computer and on mobile devices. You can visit it here:
https://randomscriptureverse.com/
Features
1) Displays a random verse from a
selection of Standard Works (you can select whichever ones you want)
2) Displays a random verse from a
specific Standard Work
3) Displays a random verse from a
book of scripture (e.g. Matthew, Alma, Moses, etc.)
4) Displays a random verse from your
Favorites (which you can manage by signing up for an account)
Ways of Randomizing
[You can select which one by clicking the settings icon near the bottom of the page, default is weighted]
1) Weighted Random first selects a random work, then book, then chapter, then verse
2) Pure Random gives a completely random verse from the pool of all possible verses
Link to LDS.org
Every random scripture comes with a URL to link to that verse in context on LDS.org (or if on a mobile device with Gospel Library downloaded, Gospel Library opens). This allows you to read the surrounding verses like you would do if opening paper scriptures.
I hope you enjoy it! I know I sure have!
-Liz Kazandzhy