I dunno, I kinda have a feeling that "illegitimacy" may not carry the same stigma for the children that it did/does in many real world societies, seeing as how a good deal of that is tied into patriarchal traditions and hangups about men and women and six that don't seem to exist in a less sixist culture like Skyrim's. But I do think that there was something about the relationship between that kid's mother and the Jarl that wasn't quite... right. Either he's the result of an affair Balgruuf had while his wife was still around, or his mother was married to someone else, or something. It seems clear from the quest that involves him that there are family issues and he brings up the "different mother" thing too quickly for it to not be a factor at least in his own mind. I can see where something happened that may have caused a lot of head-shaking and gossip and bad blood and even if it's not directed at the kid himself, kids pick up on that sort of thing and know that they figure into it somehow.
At the very least, Balgruuf has kids by two different women (that we know of) and *neither* of those women is still around in any "official" capacity. Did they both just die of natural causes? I know it's a rough world but they presumably would have had all the advantages that wealth and position could provide, unless the Jarl is a complete ass or they had a good reason NOT to stick around and make use of them. I dunno.
I was using the word "illegitimate" instead of trying to find the right English words... Born outside the marriage would have been more neutral, I guess, since I agree that it's probably not that stigmatized in Skyrim. Anyway, my private theory is that Nelkir's mother for some reason couldn't take care of him - either she died giving birth or early on in his life, or maybe she simply was poor, sick or already married (or all of it) and felt she couldn't take care of the child, here you go, Balgruuf, now you'll reap what you've sown! Likely the mother of the older kids - who I presume were married to Balgruuf - was still around. I mean, Nelkir must have believed he had the same mother as the others until Mephala told him otherwise, implying she must have been alive and around when he was born.
Now I am picturing him laying on the pick up lines:
"How do like your eggs in the morning: scrambled, or fertilized?"
"Those clothes and my clothes would look good on my floor."
"You look cold. Want to use me as a blanket?"
"I'm not actually this tall. I'm sitting on my wallet."
"Have you got any Nord in you? Would you like some?"
(from the balcony:) "See those two rutting deer out there? That's you and me, later."
"I'd sure like to climb your 7000 steps."

I think Balgruuf's gonna get some, with you two as his advisors!

Ulfric can keep his axe!
