The question at hand seems to be what would constitute legal evidence in a court proceeding of paternity. And that seems to go back to one of the key misconceptions about this whole issue. The “birther” side (term used solely for characterization, not as a pejorative) has consistently advanced the thought that mere suspicion of Obama’s birth, absent any evidence or an actual case, mandates a court determination. That is, of course, not true.
Courts don't care what ungrounded suspicions some person has about another person, nor should they. Neither will a court accept for consideration the question of anyone’s birth/paternity/etc. unless it is relevant to an actual legal case that has passed the procedural hurdles necessary to make the docket. So asking what constitutes acceptable proof of paternity in a court case absent a specific case is a bit of a non-sequitur.
In general, however, if a valid case has been accepted by a Court that requires documentation of paternity, a certified birth certificate would be the standard. It can certainly get more complicated than that if two different men who were both intimate with the mother are claiming to be the father, but that's not the case here. If Obama were in an actual court case that needed to document his paternity, a certified COLB from Hawaii would be adequate to do so. And if some unrelated people in the Court started saying they didn't believe it, they would be admonished to be quiet. If they did not do so, they would be physically removed. Their uninformed speculations would be irrelevant to any determination.