It wasn't between his children though, I don't think. One claimant was his uncle and the other was his brother, right?
Well, chaos did not erupt for almost a generation after Muhammad's death, so it is questionable to even talk about "claimants" in relation to him.
His immediate successor was Abu Bakr, his father-in-law*, followed by Umar and Uthman, both distant relatives of the same larger tribal group with hardly any sort of claim based upon family relatedness to the Prophet. In both cases, their selection was based upon widely acknowledged merit, which includes a lot of closeness to Muhammad. Uthman could also boast a pretty impressive network of tribal connection, although those connections were mostly to people that had been
opposed to Muhammad during most of his lifetime. Uthman style of rule, however, discontented a lot of people. He was assassinated. It seems that it was at
this point that conflict within the community arose. Ali, Muhammad's cousin and son-in-law, was Uthman's successor, but his legitimacy was tarred in the eyes of many by the fact that he had taken the position as a consequence of assassination, and that he refused to prosecute Uthman's killers (as he probably agreed with their grievances, if not their acts). Thus, the conflict arose between Ali's supporters and the family of 'Uthman, which,
as a family had not claim on the Caliphate as a result of a relatedness to the Prophet. Other claimants were the Zubayrids, who likewise never based their claim upon family links.
Point is, the conflict was not just about
who should succeed the Prophet, but more about
what kind of succession.
The claim from the line of Muhammad's uncle al-'Abbas emerged somewhat later. It began
in alliance with would-be Shiites as an attempt to set forth a unified claim in the name of the Family of the Prophet, except that, by that point, the different lines in that family had different claimants and agendas.
I will also note that Muhammad's wives, of many of them at least, were not at all stay-at-home types without influence on his action and message. This is critically notorious of Khadija. Another of Muhammad's wives, A'isha, famously rode a camel in battle. They are also recorded as critically important sources of transmission of Hadith.
*His accession seems to have been acquiesced by Ali at the time, although later reports, written when Shiite doctrine had given Ali a more central place, regard his rule as illegitimate in a Shiite perspective.