Supposing you are holding a pen statically. The weight of the pen is acting downward which is opposed equally with an opposite upward force by the friction force (F). The perpendicular force that acts across the surface area of the pen will be the normal force (N). At any given time, the friction force will be the product of the normal force and the coefficient of static friction (µ). i.e., F = N * µ.
If the coefficient of friction is high, like in a rough surface, then the normal force required will be less. If the coefficient of friction is very low, like in silky smooth surface, then the normal force will be high to compensate for the slippage.
To put it in other words, the coefficient of friction will be the ratio of the friction force to the normal force. µ = F/N (parallel force/perpendicular force)
This is true when the object is on the point of moving. But if you increase the normal force my holding the pen harder then friction doesn’t increase because if it did the pen would move upwards(as friction is opposing the weight of the pen which acts downwards)