A dozen or so private equity firms are taking a close look at investing in mortgage lending start-ups that claim they can originate and underwrite more cheaply than traditional players because of their cutting-edge technology. A handful of PE firms have already taken out their checkbooks, investing millions of dollars in lenders such as Better Mortgage and Nexera Holding LLC, which operates consumer-direct lender Newfi and a wholesaler called Bluestream. Roughly six months ago, Better received...