While there is no universal age for when one is allowed to decide when to die, there are a few organizations that do have some age restrictions such as the ability to order the PPH from them through Exit International. One has to be 50 years of age as well as of sound mind.
Then there is an age where one would be legally considered an adult, and in the US, it is age 18. At age 18, a person in the US is legally considered an adult, thus having the ability to buy tobacco products, vote, sign legal papers, enlist, and have full rights as any other legal adult and no longer considered a minor.
With that said, I personally believe that if one is of legal age, where one is classified under the law as an legal adult, able to make his/her own choices with regards to many decisions (including the one to enlist in the military, to serve one's country) and more, then he/she should be able to decide whether he/she wants to continue living or not. While there are people's lives that are terrible under the legal age, it would have to be a case by case basis and under heavier scrutiny than to simply allow someone who is a minor to gain access as easily. This is simply because during one's adolescent years, he/she is more impulsive and there is still a reasonable belief that there could be changes in his/her life that may allow him/her to change his/her mind. So for example, if a teenager (say 14 years of age) wishes to die because his/her life is miserable, I would say that he/she should give life at least some chance to improve and at least wait until age 18, this way he/she would have exhausted just about every option before finally going through. Another thing, like @Anxious_Panda said, teenagers haven't lived enough of a life to truly understand the ramifications of suicide. Most teenagers oftenly live at home and haven't been exposed to the real world as much as adults (sure there are some who have jobs and work while going to school), so their experiences in life as still immature compared to someone who has at least reached adulthood and/or of legal age. I think that once someone reaches legal age as an legal adult (age 18 in the US), they would have more rights and more freedom than the average teenager still under their parent's roof. They are (by law) treated as an adult and from that point on they do get to experience the things that most other adults do, hence they are able to then better decide whether life is for them or not. So by setting that minimum age at 18, they are able at least give adult life a chance and then with that information, they are able to decide whether life is worth it for them or not.
Of course, however, if said person was suffering immensely like having a terminal illness or even a horrible life, then there would be special consideration (through family and doctors' of course as the person is still a minor) in allowing that person to exit his/her suffering.
Note: Keep in mind that I am not saying that people shouldn't have a right to die, but when it comes to age, I do believe that there has to be a cut off/line in order to prevent abuse and/or make sure things are kept in check. Other than that, once someone is of legal age in the country that they are in (for the US, it is 18), then they should be free to make the decision.