Yahoo Inc. said Monday it will whittle down its board after completing its deal with Verizon Communications Inc., and several longtime directors, including Chief Executive Marissa Mayer and co-founder David Filo, will step down as directors.

After the sale of its core internet business, the company will change its name to Altaba Inc. from RemainCo, Yahoo said in a regulatory filing. Altaba’s remaining assets include Yahoo’s stake in Alibaba Group Holding Ltd. and Yahoo Japan. The name is a combination of the words “alternate”...