For a person to take a shine to someone else, he or she likes the other person immediately and to a great extent. It is usually used in reference to two people meeting for the first time, or the moment when they begin to get along. The verb take can be conjugated through all its forms.
There is little solid evidence, but most sources agree that this idiom is a variation of an earlier idiom to shine up to someone. This phrase meant to act in a way that will make someone else like you, as a suitor would to the person he or she is pursuing. It is still used, though it is not as popular.
Examples
Three suspects, members of UPFA’s Galle strongman Deputy Minister Muthuhettigama, were arrested but the subsequent action taken by Muthuhettigama to remove the suspects by force from police custody by marching into the police station and flaunting his bulk and his political power has not succeeded in making the voting public take a shine towards the Government. [The Sunday Times Sri Lanka]
Young, however, has an easy explanation for the universe taking a shine to his Lakers on Tuesday night. [Bleacher Report]
Henri also joined the company, but the pair pretended to be brother and sister because theatre companies often baulked at hiring husband and wife teams. The ruse was later discovered when a stage manager took a shine to Henri. [Melbourne Herald Sun]
There was a bit of tension at first, with Oscar’s boys and Chalky, who perhaps wondered why Oscar seemed to shine up to Chalky after so many years of separation. [IGN]