One certainly disturbing reason for the USA to become involved with the Boer Republics might be the vaguely (and by some people even rather concretely) envisioned notion to depart large numbers of blacks 'back' to Africa. This had been the whole idea behind Liberia. Jefferson first articulared the idea, I think, that 'slavery is wicked, but if we end it, the former slaves will surely hates us'. The solution later proposed by others, to some considerable acclaim, was to create a colony in Africa for free American blacks. liberia. But Liberia declared independence in the 1850s, and this was recognised by the USA in 1862. But then, the whole idea of 'that's sure a lot of free blacks, say!' cropped back up when abolition became an inevitability. The old idea ('just send them away') was certainly not frowned upon in learned circles. As late as 1858, such figures as Abraham frickin' Lincoln advocated for the abolution of slavery... on the condition that all blacks were deported.
One can imagine a scenario where, with Liberia gone, the USA opts to buy a modest 'colony' from the Boers, in order to dump (in theory) all its black people there. Now to be sure, this would never work. The logistics are insane. That's why it was never done in OTL (I'm rather cynically confident that it's pretty much the only reason it wasn't done). But whether it worksd or not hardly matters for this thread. The process would take decades, and would demand co-operation with the Boers.
When the British start threatening the Boers, the Boer Republics request to be placed under the protection of the USA. This is done, and deciding it's not worth the hassle, Britain decides not to have any kind of Boer War. Ties between the USA and the Boers tighten, and when the vast riches of the Boer Republics are discovered, US companies swoop in. The British will no doubt be reconsidering the option of war then, so the Boer Republics are quickly admitted as states.