Conservative Philosophers

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:
Elizabeth Anscombe
Reinhold Niebuhr
John Courtney Murray
Jacques Maritain
Alasdair MacIntyre
Pope John Paul II
Joseph Cardinal Ratzinger
C.S. Lewis
G.K. Chesterton

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.)