Bryan Caplan asks: "Can you name a post-1900 conservative economist as well-known as Milton Friedman, or a post-1900 conservative philosopher as well-known as Robert Nozick?"
I don't know whether or not he means well-known inside academia or well-known in general. I'll assume he means in general. If we interpret the words conservative and philosopher somewhat broadly, then I'll throw these names out there:
John Courtney Murray
Pope John Paul II
Joseph Cardinal Ratzinger
As for conservative economists comparable in fame to Friedman, I'll throw in the towel. However, I will say that considered himself "a libertarian with a small 'l' and a Republican with a capital 'R.'" He also called himself a classical liberal. But these descriptions are not so different from what many contemporary conservatives would apply to themselves. And maintaining a long-term connection to the Republican Party, as Friedman did, is something many libertarians would find too distasteful to attempt.
Also, Hayek, though he denied being a conservative, actually was quite conservative in his philosophical views (e.g. a reluctance to tinker with institutions which had presumably evolved that way for a reason.)