What rights does the son have?
FWIW under U.S. law:
The son presumably has dozens of rights and trying to create a list would serve no useful purpose.
You told us that the father "agree[d] to sign [and, presumably, did sign] a document removing his name from property title." You didn't say for sure, but I suppose we should assume the document he signed was a deed. You then told us that "Mum never submit[ted] this or [made] this official." I have no idea what "makes this official" might means, but I guess we're supposed to assume that this means the mother never filed or recorded the deed with whatever public records office one files/records real property records in Australia.
All correct? If so, as between the mother and father, title has been transferred. That the mother failed to file/record the deed only matters if someone acted in reliance on the title records, which doesn't appear to be the case.
You then told us that the mother and the "eldest son [entered into an oral agreement whereby the son agreed] to contribute [some unstated amount(s) of money in exchange" for 50% of future capital gain." In other words, the agreement was for the son to make financial contributions toward the upkeep of the home in exchange for the payment of money at some time in the future which isn't clear from your description. The statute of frauds does not apply to such an agreement.
Finally, you told us that the mother "die[d] [and that] estate lawyers [have] suggest[ed] that dad owns 100 percent of the property, including all capital gain ...despite not doing anything for all of those years and the son taking on the burden." For starters, that the father did not "do[] anything for all of those years" and that "the son [took] on the burden" is irrelevant to who owns the property. Based on the facts provided, the mother owned the property in its entirety at the time of the mother's death. Of course, I'm assuming that the unfiled/unrecorded deed still exists in order to prove the title transfer. That being the case, there are at least three issues:
First, what happens to the property? If the mother owned it outright at the time of her death, then title passes in accordance with her will or, if she had no will, with the provisions of the intestate succession laws of the relevant jurisdiction. You didn't tell us if she had a will or, if she did, what the will says. While I don't care to research how intestate succession works in Australia, under the law of every U.S. state, where a person is survived by a spouse and one or more children, either the spouse inherits everything or the spouse and the children divide the estate.
The second issue is based on your statement that the estate lawyers said that the father "owns 100 percent of the property,
including capital gain." That makes no sense because title to property and a capital gain are two very different things. This brings into focus the vagueness of the alleged oral agreement. You told us that the consideration for the son's contributions was "50% of future capital gain," but what does that really mean? A "capital gain" is realized when a capital asset is sold. But this property was never sold, so the mother never realized a capital gain, and, if the estate simply transfers the property to the father, then the estate never realizes a capital gain either.
The third issue is the simplest. The only person who can apparently testify about the alleged oral agreement is the son. The mother cannot do so because she's dead. Did anyone witness the making of the agreement? If the only evidence of the agreement is the son's self-serving testimony, he may not be able to meet his burden of proof.
Obviously, if this is not merely some hypothetical, the son should retain the services of an attorney.
