One of the biggest mistakes I've ever made is designing a menu structure poorly. It lead to a 100,000 downloads and counting app making a grand total of $6.
My mistake? Building a primarily multi-player game where the menu gave focus to the single player (late addition and less refined) content, made it difficult to find tutorials and makes in-app purchases unattractive and even difficult to find.
The menu was beautiful but it didn't do it's job, it didn't quickly and simply show the user what they wanted to find. It didn't lead the user where they needed to go, it didn't fast track the user to the game itself. Huge problems given minimal consideration.
So what makes a good menu structure?
Take me too the game
If it takes more than 2 taps to get from the menu to the game, it's taking too long. One game that especially impressed me was Jetpack Joyride by Halfbrick. The game has both a minimal and extended menu, the first time you play you are given 1 option "Tap anywhere to begin", you can't focus more on taking the player to the action than that!
Set the focus point
Having your important buttons bigger and and more central than the other options instantly makes the player focus on that option. If you need to have a more extensive menu structure then it guides the user through a default set of options by simply following the biggest buttons.
Develop your menus with your monetization strategy
A lot of developers favor having a separate store on their main menu where players can go through and find additional content they wish to buy. Something to consider is mixing your in-app content with the actual games content. Such as allowing players to buy "Save Me" tokens at the point of them failing in the game itself, kind of like adding coins at the continue screen of an arcade game. If you have a character select screen why not add your locked in-app purchases to the list as well.
Keep it simple
So many apps fail this step but it is by far the most important. If you don't need an option or level of menu options then cull them, cull as much as possible, give the player the bare minimum number of options. Use prototyping and design software to layout and revise your interface. Keep iterating and removing things that aren't needed. This will not only cut down on development time but make the players experience smoother.
I cannot stress enough that last point about keeping it simple, draft up your concept menu structure, revise it and work through it with a handful of different people until everyone you talk to is comfortable with it and feels that your users will be as well.
I've gone through this process thoroughly with my next upcoming (February) game and hopefully I will have some good statistics of how the experience has been improved.