First, my memories, which might not be accurate. Late in the 1980s, right before my grandmother moved out of her house to the big one on the beach. She had MTV, and even though I was only eight years old and not really interested in music yet, I was interested in MTV, because like the microwave oven, it felt opulent. And so I remember watching the video for this song over and over, and the refrain being unavoidable during one of my trips there. And maybe I even remember a comment from my mother---how the song was him trying to be a pop star again. Him being George Harrison, my mother's favorite Beatle, but at the time I didn't really even know who the Beatles were. Maybe I am confusing things, maybe the song was on the radio, maybe I didn't even watch it on MTV.
So that out of the way, let us look at the history of the song. "Got My Mind Set On You" was originally written by Rudy Clark and performed by James Ray, an R&B singer who died at the age of 22, soon after recording the song in 1962. The song, as originally performed, combined elements of R&B (like a horn section) with more straight-ahead rocking. And it was that version of the song that a young George Harrison found in a record shop while still young, before the Beatles blew up. And then George Harrison was a Beatle, and then during the 1970s he embraced his own style, with many songs and lyrics focusing on spirituality.
In 1987, after some time away from recording, George Harrison, together with producer Jeff Lynne, released Cloud Nine, a popular album. And the first and most successful single off of it was "Got My Mind Set On You", which reached #1 in the charts in the US (although only #2 in Great Britain), making it his first #1 single since 1973 and Give Me Love (Give Me Peace On Earth). The song was a simple and repetitive love song that even seems a little crass, as the singer repeatedly states that it will take time and money to win a girl's affections. And while very catchy, like many catchy songs, it can quickly become irritating. It was also a departure from Harrison's spiritually-oriented material of the 1970s.
While it might seem like the move from spiritual, mature songs to a simplistic song like this was George Harrison attempting to make a pop hit for money and/or fame, I think it is just as likely that Harrison was just having fun. Before he was an activist and visionary solo artist, and before he was a Beatle, Harrison was just a young man playing R&B records for fun, and I think this recording was a way for him to recapture his first musical influences.