Guys
I think emigration was already fairly large at the time and the population had pretty much peaked for the technology and organisation of the time. Hence, if somehow the famine had been largely avoided then you would very likely see large scale emigration from Ireland to Britain and wider afield in a generation or two as transport systems increase and become cheaper. However it would probably remove a lot of the venom that poisoned relations. [Probably the easiest butterfly would be change the timing so the Tories are in charge for the duration, with a more hands on approach rather than leave it to laissez-faire as the Liberals did. Not sure whether this would be able to extend for the duration of the famine but could help greatly]. Also some better communications so the details are known earlier. [One thing I once read was that in the early stages questions were asked about the famine and the blight but Dublin reported no problem. Basically because the potatoes around there were fine. However the slightly different type in the much wetter west and south was already rotting in the fields].
If relations are better and there is less tension there will still be problems as at least some of the Catholic groups will start arguing for restoring a Dublin Parliament and the Unionists and their supporters in Britain will oppose this. Likely to be a fair bit of stress on this but don't know whether a solution can be provided without the OTL conflict? Would it be possible to get an agreement similar to OTL without the violence, i.e. Dublin has Home Rule from London and Belfast from Dublin?
The other option, although I don't think there's much in the way of raw materials for it, would be if there was a markedly greater level of industrialisation in Ireland itself, to generate wealth and make use of the manpower and market coming from the larger population. Again this would need Britain to avoid the laissez-faire policy, but then in this time period that would be good for Britain as a whole.
Given the anti-Catholic feeling in the US and the occasional tension between Britain and America it might well be that an higher proportion of the Catholic diaspora goes to other areas, although that of course needs the stability and economic development to mean there is a pull as well as a push factor. Baring fairly dramatic changes I think the most likely locations would be what became the settler dominions and Argentina but how many more would go to them and what the impacts would be is hard to tell.
Anyway, my initial musings on the idea. For what their worth.
Steve