I like when remixes have a balance of the original work and the remixed work. I don't like it when, let's say, a drum loop is changed and the rest stays the same and I find it really weird when A) the remixer just KILLS the song by forcing his/her style into the song (check Tiesto's shitty Clarity remix) or B) the remix is, although good, a completely new thing.
My best advice is
1.- Pick a song that YOU LIKE and that you know YOU CAN ADD TO. I once had this idea of "Imma remix all the big names to get all the attention and the moneyz" but it is very difficult to remix, let's say, an Aphex Twin song if you were never into Aphex Twin.
2.- Appropiate the song. This basically means that you give it emotion and meaning, so that your work isn't just a remix but a different perspective on art itself.
3.- Know the music key and the bpm. If you want to go the next level, if the key is minor and you want to make the song happy, try writing music in its relative major and viceversa.
4.- Pay close attention to the hooks or iconic elements of the song: the lead, the bassline, the vocals, the drumloop even just the pads. Have them in your remix but think of what can change around them. E.g. a very nice lead in a song can have a different pad around it, or be played in a different instrument.
5.- Try to improvise music around it. If you can play piano or guitar, listen to the song and jam along it. Even if you repeat the same chords or melody over and over again, it is great to take out ideas.
6.- Focus, at the start, much more on the chorus/drop rather than intro>silence>pre-chorus>chorus/drop>silence and that scheme.
7.- It's cool to remember that less is more. I love to go batshit crazy and try to add as much of my work on a remix to make the track shine, but it is sometimes too much. Again, think what you can add and what vibe can you change.
Here are some personal reccomendations that have that original work-great ideas from the remixes balance. Good luck

Bob Sinclar's Remix of Music Sounds Better With You by Stardust
Chrome Spark's Remix of Goodbye To A World by Porter Robinson
Robotaki's Remix of A King Alone by The M Machine
Syn Cole's Remix of Silhouettes by Avicii.