What is the best way to rotate a CGPoint on a grid?

Swift 5

extension CGPoint {
    func rotate(around center: CGPoint, angle: CGFloat) -> CGPoint {
        let translate = CGAffineTransform(translationX: -center.x, y: -center.y)
        let transform = translate.concatenating(CGAffineTransform(rotationAngle: angle))
        let rotated = applying(transform)
        return rotated.applying(CGAffineTransform(translationX: center.x, y: center.y))
    }
}

Use a CGAffineTransform.

You can also use that:

rotatedPoint = CGPointApplyAffineTransform(initialPoint, CGAffineTransformMakeRotation(angle));

EDIT: to perform rotation around custom point you must do like Adam described in his answer. Using CGAffineTransform it must look something like:

CGAffineTransform translateTransform = CGAffineTransformMakeTranslation(customCenter.x, customCenter.y);
CGAffineTransform rotationTransform = CGAffineTransformMakeRotation(angle);

CGAffineTransform customRotation = CGAffineTransformConcat(CGAffineTransformConcat( CGAffineTransformInvert(translateTransform), rotationTransform), translateTransform);

rotatedPoint = CGPointApplyAffineTransform(initialPoint, customRotation);

Use a 2D rotation matrix. If you want to rotate a point counterclockwise about the origin by an angle of angle, then you would do this:

CGPoint RotatePointAboutOrigin(CGPoint point, float angle)
{
    float s = sinf(angle);
    float c = cosf(angle);
    return CGPointMake(c * point.x - s * point.y, s * point.x + c * point.y);
}

If you want to rotate about a point other than the origin, you'll have to first subtract the center of rotation from your point, rotate it using the above, and then add back in the center of rotation (this is called conjugation in matrix theory).