It seems that the Home Secretary Lord Phillips was referring to in that interview was Charles Clarke, who was Home Secretary from December 2004 to May 2006.
Lord Bingham was Senior Lord of Appeal in Ordinary from 6 June 2000 to 30 September 2008. Over that period, there were five different holders of the office of Home Secretary:
Given Lord Phillips' use of the male pronoun, it seemed unlikely that he was referring to Jacqui Smith. A few Google searches later using combinations of search terms similar to:
"Lord Phillips" "Lord Bingham" "Charles Clarke" "Law Lords" discuss
(just modifying the search to include each Home Secretary in turn) I found links to another occasion where Lord Phillips talked about those events, and on that occasion he mentioned the Home Secretary by name.
Lord Phillips mentioned Charles Clarke's request for a meeting with the Law Lords, and the fact that Lord Bingham had declined the request, in a speech titled “Judicial Independence”, which he gave at the Commonwealth Law Conference in 2007:
Charles Clarke, when he was Home Secretary, was keen to discuss with the Law Lords, in advance of taking such measures, issues of principle that they might raise and he was aggrieved when the Senior Law Lord, Lord Bingham, declined an invitation to meet.
The context is important here. In the speech at the Commonwealth Law Conference in 2007, Lord Phillips went on to explain why Lord Bingham declined the invitation to meet with the Home Secretary:
"... it is the judge’s job to resolve disputes as to the legality of action or legislation when those disputes arise. If he advises the Government on that question beforehand he will place himself in a position where he cannot do so, or
appear to do so, impartially. The House of Lords Select Committee commented:
‘It is essential that the Law Lords, as the court of last resort, should not even be perceived to have prejudged an issue as a result of communications with