There are minhagim that state one should keep a candle lit during the entire Shiva. There is also a Minhag which I see many here observe, to keep it going for an entire year. There are few mekoros for this, although I believe it’s mentioned in Nitei Gavriel. a) what is the status of this minhag? b) what if children of aveylim say they would like to keep such a minhag, is it bad (or a false minhag) that one should not undertake?

In regard to minhagim, what determines what you should do is your family or community minhag. The fact that you have seen people abiding by a certain practice and people you know began to follow them does not require you to do so. Today, many minhagim of different communities appear in sefarim for everybody to see, and as much as people may feel that they are missing out on a beautiful minhag, it does not require them to do so, unless it is an explicit halacha which one was unaware of. On the other hand, if one practiced a certain way because he thought it was an obligation, but then realized that he was not required to do so, he may discontinue that minhag even without "hatarat nedarim'. In regard to your specific question of lighting a candle for the entire year, I found this minhag mentioned in "Tzava'at Rav Naftali Katz zt"l"(סעיף כד). (Also in Mourning in Halacha pg. 179 note 7) His request was though, that a candle be lit in the Bet Knesset not in his home. I suggest though reading the Tzava'ah and you will realize that Rav Naftali Katz zt"l was a unique individual who asked for many special things to be done after he was niftar. [You can read the sefer online at http://www.hebrewbooks.org/20832 . ] So, since we found it was asked by a great Rav, we can see it as a kosher minhag. However, as I said before it is not binding. One should bear in mind though, that the greatest thing to be done for the neshama of the Niftar is to learn Torah. (See Kiddushin 31 that one is required to respect his father even after his death and קב הישר פרק ל"ה, נטעי גבריאל הלכות אבילות ח"ב עמ' תסו-תסח) In general the local chevra kadisha is most up to date on the local minhagim. Besorot tovot and Happy Purim