Making the choice should be as fast as possible. I can't remember why, but I know giving the user the fastest way to make a choice (by narrowing down their available choice pool) will lead to a better user experience.
EDIT: As I explained in @Kevin's answer, it's because of information overload. The options placed at the top ("15min", "30min", etc.) are there so as to make it easier for the user to guess. But @Kevin had a good point about culture, if your app is going to be used by people from different countries, make certain you take that into consideration.
With that said, you probably want to use Option 1 (I prefer it to Option 2), but with some buttons at the top of it:
The time range below should of course update to match the button selected.
This will work well with most users, since they probably want to make the choice as quickly as possible. Giving the user only the option of picking the time range could lead to information overload. Using only buttons leads to less mental work, and this will lead to faster choice. Which is good, i guess.
This might be one of the reasons why YouTube uses this horizontal bars instead of a slider to change the video's speed:
Additionally, you can use some machine learning to tailor the experience to the user. If the user keeps picking 15mins then make sure that the 15mins button is already selected. If the user keeps picking a certain time range (e.g. from 02:00AM to 04:00AM), then have the app respond accordingly.