I have two ideas that haven't been tethered to a world yet:

1a. This magic is like FMA alchemy or feng shui. Magical effects are created by manipulating the flow of chi/prana/"energy". This can be done through drawn symbols, strangely shaped musical or mechanical instruments, dance, etc. This is a science and 2-dimensional symbols are the hardest to perfect. These methods must also be tailored to the environment they are being used in, since the energy flows differently in different areas.

1b. In the same world there are an extreme minority of people who break the conservation of energy. These people are metaphysically tethered to higher dimensional beings who release or absorb energy through their vessels in the mortal world. Because it's very hard to alter the body in a way that is sufficiently different and doesn't kill, people in this situation only have the ability to use one power. Despite this limitation (and any mental instability associated with being mind-mates with Yog-Sothoth) their powers are much stronger than most veteran alchemists/"scientists".

1. This idea is really more of a synthesis of the above two. Magic is a hidden system designed for an extremely alien species, though it follows the same principles as fantasy magic. There are magic words but they're not pronounceable by humans. The sounds are at octaves too high or too low for human vocal chords and ears and the vowels and consonants themselves cannot be produced by a human mouth. There are magic gestures but they require additional arms, joints, and movements that would put any single human in the hospital for a while if successfully performed. There are magical reagents but they're made from rare metals and strange compounds.

This magical system was discovered in an industrial accident. Three workers in a busy manufacturing plant collided with each other while carrying various materials. As they spilled their supplies and flailed about they opened a rift to another world, which lasted only a few minutes. Over years of experimentation scientists were able to replicate the approximate forms and voices of the alien magic users. Magic users walk around with mechanical apparatuses looking like one-man bands, playing songs no one can hear and dancing like electrocuted insects.

...of course they never planned on meeting the aliens themselves, who have given up their heathen magic in the name of Science! and wish to eliminate its influence from the universe. cue drama