One of the most effective ways to interpret a popular song on piano is to see what musical styles are implied in the original version, and then magnify them.
When I set out to play The Beatles "In My Life" for my Complete Beatles Piano video series on YouTube, I expected to play the song like I've always played it before: as a nice, moderate pop tune with a semi-classical interlude. But as I practiced it and envisioned the video, I kept exploring the classical influence in the interlude, and eventually decided to play the whole song in the style of Mozart.
Some of it sounds very much like Mozart, and some less so. But that's the point. Magnifying the classical influence enabled me to personalize my interpretation in a way that I hadn't thought of before.
You can hear it here:
This is only just one possibility. I could have gone in the opposite direction, by highlighting the pop groove and even playing the classical part with the pop beat underneath. Solo piano playing offers us pianists a great deal of interpretive freedom, and magnifying musical implications is one way to explore a song's stylistic possibilities. And since the Beatles hinted as so many different styles of music themselves, they open up a whole universe of solo piano choices for us to dive into.
Good luck with your music, and see if you can practice these concepts on a song of your choice.