There are two types of people who get into mobile gaming: those who have long periods of down time and those who only have a few moments. While these two people have the same need for mobile gaming, they have vastly different needs when it comes to the type of software they buy.
If you're the type of mobile gamer who has long periods of free time, then you're the lucky type because you're choices are not limited. You can either play games that require your attention for long periods of time, or you can play games that are broken up into smaller, quicker chunks. Depending on your budget, you may even want to think about getting a portable device such as a Nintendo DS or a Sony PSP.
If you're the type of mobile gamer who only has short but fairly frequent periods of free time, then you're a bit out of luck. You're going to be more limited in your game choices but if done correctly you can also enjoy your gaming.
A lot of people go with more complicated games thinking that it will be worth their buy more, but these games usually demand more time and if you don't have the time they may start to feel frustrating because you rarely get to finish a section of the game. Instead, go for such games as Monopoly, Bejewled, Chess, and maybe a turn-based RPG such as Orcs & Elves. This is because you can take a turn, make a move, or take a few actions and pause the game at one if its natural breaking points. The more complicated games can also be paused, but it will most likely be at random points, making the experience feel broken, it breaks the flow of the game, and this may suck the fun out of it.
Being a mobile gamer does not mean playing boring games. All mobile games can be fun, you just have to choose the right game for you.
Avoiding movie tie in games are often a good bet. The the 24 game that came out a few years ago was infact, pretty righteous. Easy controls, and was pretty forgiving when quick action was needed.
Its all very hit or miss. I must say, though, that DoomRPG was a very solid experience and did a decent job of saving progress very close to the point where you stopped playing.