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<blockquote>If you can do it with your mind, then why can't everyone do it, and why are there wizards with inherently different strengths?</blockquote>

It is like many other abilities we are already familiar with.  Some people are physically stronger than others, or faster, or have better hearing, or are smarter.

Then there is training.  While some people were born with potential to be exceptionally strong, they still have to do lots of workouts and training to realize that potential.  Not everyone can be an olympic gold medalist sprinter no matter how hard they train, but all gold medalist sprinters had to train hard to get there.

Your exceptional wizards not only have to be born with the right genes, but have to be taught to harness the power, and have to do lots of practice to  use it effectively.  If you want to be really good, you have to dedicate your life to it.

<blockquote>Any magic should come with a cost (or else it would be overpowered). What would be a reasonable cost for this type of ability?</blockquote>

There are lots of ways this could go.

It could be physically draining, like sprinting up a hill.  The more improbable, the more draining it is.  Using the power diminishes the ability to use the power until you've rested.  Using it too much without sufficient resting could have long term detrimental effects.

It could also require stealing resources from other biological functions.  For example, it may rapidly deplete the blood oxygen level.  If you use the power too much without taking time to recover, you pass out.  Do that too often, and you suffer long term brain damage.  Again, the more improbable the more oxygen is taken from the blood.

Similarly, it could require a particular scarce nutrient in the environment.  Imagine if you had to eat a gram of saffron to pull off a meaningful trick.  Or, it could take a bit of stuff called "spice" that turns out to be worm poop from a distant planet that is heavily guarded and expensive to get to (Nah, that's silly, forget that one).  

Another possibility is that there is a long term cost for each usage of the power.  Your body is damaged a little be with every use.  Some people are more susceptible than others.  Think of it like having to smoke one cigarette for little feats, and multiple packs for large feats.  Some people do it and get away with it and live long lives.  Others die from an incurable disease much more quickly.  You don't know about any one individual, but on average, there is a clear downside.