What is a minhag? "Custom." Plural: minhagim. However, some minhagim have been practiced so widely and for so long that rabbis have declared them to be mandatory halacha. More generally, each large group of Judaism are considered to have a "minhag," including Ashkenazi, Sephardi, Mizrahi, Yemenite, Indian (from India), etc.
Further, some minhagim are how someone does an actual halacha. One of the most common examples is how a person ritually washes his or her hands. There is no wrong answer. You're fulfilling the mitzvah, but there are several customs on how to fulfill it.
Back to the point at hand. This is a question that born-Jews ask me all the time. As far as I can tell, there are basically four sources that seem to form a hierarchy:
1) The first one you hear/learn.
2) A significant other's minhag.
3) Your community's minhag.
4) The one you like better.
Of course, some people may skip directly to the personal preference. Interestingly, born-Jews constantly encourage me to pick the "most convenient" option and express jealousy that I get to choose!
Let's be clear that I don't mean Ashkenazi v. Sephardi. As a general rule, you will have to choose there. And as nearly no Sephardi community does conversions, chances are that you should consider yourself Ashkenazi. However, my understanding is that you may change that self-designation later, particularly if your heritage is a Sephardi culture. The only exception here is Hebrew pronunciation. Many Ashkenazi baalei teshuva and converts choose to use modern Sephardi pronunciation because of Israel. (However, note that there is a real push for Jews of color to "be" Sephardi. Interestingly, I've seen this peer pressure even between Jews of color, so it's not just rabbinic/born-Jew racism/stereotyping.)
Don't forget that some of this choosing may be irrelevant. If a convert marries a born-Jew, there is certainly a temptation to adopt the minhagim of the spouse and the spouse's family. However, for females, it is tradition (halacha?) for a wife to adopt the minhagim of her husband. Being a female convert, I look forward to following both of those reasons. Unless he waits 6 hours between meat and dairy. (For convenience and health reasons, I chose 3 hours as my minhag!)
The lesson here is that you may adopt whichever minhagim you wish, regardless of what someone may tell you. As long as you are within acceptable custom and halacha, you may have a mishmash of minhagim; and even for born-Jews, a mishmash is becoming much more common. There is no "right" minhag, even though inevitably, someone will tell you that a minhag is "pure halacha." Personally, and because of practicality, most of my minhagim are the first ones I learned. It was just simpler that way.
The only problem with that method is that I don't always remember where my customs came from. That's a problem because I can't explain it or track down a reasoning whenever someone tells me I'm practicing some strict minority opinion that very few people follow. Then I'm stuck giving them an awkward smile and "ummm...it's just what I know." Then either a lecture happens or the person walks away thinking I'm a nut.