Not just the herpes scare. I turned 18 in 1985, and was still a virgin, mostly because of HIV. I wanted to be a full grown adult able to get my own condoms, medical advice, etc.-- plus the idea of telling the four parents in my life I had HIV or was pregnant was more than I could stomach.
Anyway, for relative DNA matches, you want, ideally for a man, a male relative on the paternal side. For the paternal side, they match the Y-chromosome, which is very slow-mutating so that related men have pretty much have the same Y-chromosome. FWIW, in establishing maternal relationships, they match mitochondrial DNA, which is inherited only from the mother, and is also very slow-mutating.
Chelsea would have neither a Y, nor mitochondrial DNA in common with her father, but her DNA could still be useful in showing a familial relationship, it would just have to be a more complex test. Matching Ys or mDNA are just easier tests. Like someone said, though, there’s always the possibility he didn’t father her. She looks too much like him, and I’m pretty sure he’s her father, but for legal reasons, I’ll bet her DNA would not be enough to fully establish paternity.
If he and his brother share the same father, you can match the Y, but if they share the same mother, you are out of luck: different Ys, and even though the brothers share mDNA, Clinton wouldn’t be expected to share it with the child in question.
A half-uncle would share some DNA, but I’m not sure how much, and whether it’s enough to rule Clinton in to the exclusion of enough other men to order a test on him.
This is the legal thicket: ultimately, you do need a test of Bill Clinton’s own DNA. A match that says Chelsea is the person’s half-sister is probably good enough to get a court order requiring Clinton to submit to a DNA test. A test from a half-brother with the same Y-chromosome that showed a Y-chromosome match (but that ruled out that brother himself) would probably also be enough, but something saying simply that the half-brother could not be excluded as a relative, without the Y match, might not be enough to compel a test from Clinton.
We don’t know what the mother has to say. Sometimes the mother’s affidavit is compelling all by itself (“He has a birthmark that looks like Florida on his junk”), and a judge will order the test based on that alone. Sometimes the alleged father has rebuttal evidence (“I had the flu the week she claims we took a road trip together, and here is a letter from my doctor at the time, and photocopies from my records”). This is where partial matches usually tip the scale in favor of getting the test done.
Disclaimer: IANAL, but I have helped people wade through the legal waters of establishing paternity when I did social work.
I see someone else has already answered the question of whether the man in question is Clinton’s son. I still wanted to go ahead and address the original query regarding what would be enough evidence to prove paternity.