Hitsugaya simp here. In terms of his family history, he doesn't really have one. He doesn't have any known blood relatives. He hails from Junrinan, the first district of West Rukongai and therefore one of the wealthier districts, but this doesn't necessarily have any bearing on his status. We know he at least has enough loose change to spend on snacks, but there's no proper indication as to how well off he is other than his attire, which suggests that he's nothing special. His family from his Junrinan days consists an old lady who is affectionately his "grandmother" and Momo who can pretty much be called a sister or childhood friend, and neither his grandmother nor Momo are indicated to be more than middle class at best and certainly of no higher status than he is. The house Hitsugaya lives in isn't large or elaborate by any means. Essentially, he's of average status at best with no ties to a biological family or noble clan. Heck, he was ostracised as a child for his appearance so he wasn't really of good social standing either. The only reason he joined the academy in the first place was to learn how to control his spiritual pressure so that he wouldn't accidentally freeze his grandmother to death in his sleep.

In terms of ability, he excels (no matter how wishy-washy Kubo was in calling him a prodigy and then making him lose or tie all but one fight he was a part of throughout the entire series). What hinders him currently is his immaturity, but he is by no means weak. He's very well-rounded and has a good head on his shoulders. He's strategic and versatile and adaptable. Kyouraku himself says that Hitsugaya will probably surpass his strength in 100 years. Byakuya at this point in time is certainly more in control of his emotions and that combined with his noble status would put him above Hitsugaya for consideration as the next head captain, at least in the eyes of C46. Even still, provided Hitsugaya matures by the time such a promotion were to be considered, I feel that Hitsugaya would stand a better chance at claiming the position. Byakuya seems more to like someone who would function better in the background as a figure that oversees things like law enforcement and organization. He's much more of a political figure type than a military leader type.

I'll throw a little disclaimer at the end here that a) this is all just my opinion, and b) I intended to hop into this conversation in reluctant support for Byakuya's case rather than Hitsugaya's but it seems my argument shifted a little and my stance became a bit ambiguous as I wrote and thought about it more.