Although no MP went on record favouring a threshold, when the European Referendum Act passed to the House of Lords, the issue of a threshold was discussed.
Firstly, on October 13th, 2015, the issue of a threshold was discussed, in particular by Lord Wallace of Saltaire, who noted in his contribution:
The noble Lord, Lord Norton, and other noble Lords raised the question
of threshold, which clearly we will have to explore a little, although
it is a very difficult issue. Whatever happens at the end of it, if we
have a narrow majority, either with a low or a high turnout, it will
not settle the issue. However, we all know that referendums do not
settle the issue.
This seems to imply that he supported a higher threshold than 50%. Later on, on November 4th, Lord Kerr of Kinlochard noted:
It is important to note that we have no threshold requirements in this
referendum and we have had no amendment in Committee proposing that
there should be a threshold. That is constitutionally quite surprising
for a decision as big as this. The precedent would lead one to think
about a threshold. I would not have wanted a threshold. I would not
have wanted a supermajority, as in the precedent in Scotland in the
1970s. I do not like referenda but the essence of a referendum is that
you win or lose. It is clean; it is 51% to 49%, for example. If 51%
are in favour of our leaving the European Union, we will leave, and we
should not create any fudge round that. This is a yes/no decision,
and if you decide to go, you go. The double referenda theory
attributed to Boris Johnson, which he appears to have come off—that if
the decision was to go, there would be another negotiation in which
the foreigners, astonished and timorous, would come creeping, offering
us far better terms to stay in—is nonsense. If the country votes to
leave the Government will be required to invoke Article 50 and start
the process of coming out.
So Lord Kerr would not have supported a higher threshold or supermajority, and no amendment from either House attempted to build one into the Act.
The main attempt to build a threshold of sorts into the Act was the attempt by the SNP in the Commons (see here, amendment NC3) to ensure that every constituent country would individually have to return a majority vote in order for the referendum to advise the government to leave the EU. This was later also proposed in the Lords by Lord Wrigley. Neither of these amendments succeded.
