Bookmaker William Hill twice shortened their odds for Davidson to be next leader of the Conservative party during the debate, first from 33/1 to 25/1 and then to 16/1.
"Ruth Davidson made a big impact, with a significant flurry of support for her throughout the debate to be the next Tory leader," said William Hill's spokesman Graham Sharpe.
"And while we took plenty of bets on the outcome of the referendum during the debate – 70 per cent of them for Leave, they were of moderate size and we have not changed the odds from 2/9 Remain, 3/1 Leave."
Boris Johnson is still favourite to take over from Cameron at 5/2, though his odds would undoubtedly shift out if the UK votes to Remain.
Davidson recently led the Conservatives to second place in the Scottish parliamentary election, overtaking Labour, while the Scottish National Party (who strongly back Remain) dominated.
Meanwhile at the debate, Johnson and current London mayor Sadiq Khan exchanged heated words.