Nobusuke KISHI was also Ex-Prime Minister. (I missed it) So If Hatoyama becomes the PM(and I think he will surely do), we will have 4 PMs from ex-PM's familywas consecutively.

Hobby, Most simple explanation is that he has the right wing ideology. But I think more important factor for him is his constituency. His father was an ex-health minsiter, which is the position in charge of the pension fund for the bereaved of the solders who died in the last war. The family association of the bereaved of soldiers, who are the beneficiary of the fund, was an important political base for Kozumi's clan. And this association is the strongest advocate to pressure politicians to pay homage in Yasukuni shrine on the August 15 memorial day. Koizumi promised with them to go to Yasukuni on the day as the PM, (usually PMs in Japan do not go to Yasukuni on that symbolic day to avoid the contraversy) before he was elected as the PM. He kept that promise to court their constituency. This type of attitude (loyalty issue) usually receives both positive and negative responses from the people in domestic politics in Japan. In his case it contributed to his popularity.