Traditionally, the way to become a Jew is either to be born to a Jewish mother or to convert. However, two branches of Judaism--Reform and Reconstructionist--in some cases recognize that an individual with a Jewish father but not a Jewish mother can also be considered a Jew without formally converting to Judaism.
So this issue cuts to the heart of an eternal question in Judaism: who is a Jew? How does one become a member of the people Israel?
Well, we all agree that if you have a Jewish mother, you're a Jew. Even if you don't know the first thing about Judaism, if you're a guy with a Jewish mother, you could go to any synagogue tomorrow and count as part of a minyan--a quorum. (You may not ever choose to, of course, but you could.)
But what about the guy who's mother is not Jewish, but who's father is? Let's say the parents chose to raise this guy Jewish. Should he have to formally convert to Judaism?
The traditional answer is yes. It has to come through your mother--if your mother's not Jewish, and you want to be Jewish, that's fine but you have to convert.
I suppose you can argue that if the guy does formally convert, he'll make everyone happy and there won't be an issue. Except that he won't really make everyone happy--because not everyone agrees on what constitutes a valid conversion.
The Orthodox will say that you must have an Orthodox conversion in order to be 'really' Jewish. Conservative Jews say that the conversion must be Orthodox or Conservative. Reform and Reconstructionist Jews generally accept conversions to any branch as valid. The state of Israel, as I understand the current situation, says only certain Orthodox conversions are valid.
Oy!

This whole question, I suppose, gets into issues about what constitutes a valid conversion. Orthodox and Conservative synagogues insist on intense study, appearance before a beit din (a sort of small, rabbinic court), a mikvah (ritual bath) and circumcision for a male (or a symbolic pricking of blood for those already circumcised).
Reform and Reconstructionist synagogues don't have all those requirements. (Or don't always have all those requirements.) They may require intense study and appearance before a beit din, for example, but not a mikvah and, in some Reform cases, at least, not circumcision.
Long story short: if the guy who identifies as a Jew based on patrilineal descent agrees to convert to 'make everybody happy'--well, he's not likely to make everybody happy anyway. Not unless he converts with just the right Orthodox rabbi.
Does he need to make everybody happy? No. (And this is why most converts to Reform, Reconstructionist or Conservative Judaism don't bother getting an Orthodox conversion.) I mean, if this theoretical fellow is satisfied in his Reform community, the issue will never come up. It's only going to come up if, say, he marries a girl who's Orthodox or Conservative. Or, I suppose, if he has Orthodox or Conservative relations who are on his case.
Patralineal descent is an issue within my family. I've always solved it for myself by considering a Jew to be anyone who would be accepted as such by one of the four major branches of Judaism: Orthodox, Conservative, Reform or Reconstructionist. So that means that I do accept patrilineal descent.
Judaism's hard enough--do we really have to make it harder on the people who want to identify as a Jew based on their father's Judaism?
