Yeah, Spider isn't an insect totem, but it does have some... interesting... motivations.
At least one of the Sam Verner novels deals with Spider as a totem, and the results are frightening, to say the least. Still, I would totally allow it as a PC Mentor Spirit in my game.
Bug City from 2nd, and Magic In The Shadows from 3rd both lists Spider as preferring quiet, dark places and disliking open areas away from shelter, and as a master weaver who waits patiently for all things to come to her. In those editions the benefits are related to illusion spells and nature spirits, while the disadvantages are related to being out in the open or not having time to plan.
Street Magic from 4th expands on this by giving bonuses to Detection and Illusion spells, and requiring a composure test (3) to make quick decisions in situations they haven't prepared for.
With that in mind, I would stat Spider as follows for SR5:
SPIDER
Spider weaves the web of worlds and waits patiently for all things to come to her. Her mood may be dark and fearsome, but she takes a curious interest in her followers and will protect them behind her web, ruthlessly capturing invaders who might do them harm. From the center of her web she is the mistress of schemes and conspiracies, but she does not appreciate surprises that manage to slip past her silken strands.
Advantages
All: +2 dice pool modifier for either Assensing or Leadership Tests
Magician: +2 dice to either Detection or Illusion spells, preparations, and rituals
Adept: 1 free level of Commanding Voice Authoritative Tone, or free Motion Sense or Three-Dimensional Memory
Disadvantages
Spider magicians are not quick on their feet. They must make a Willpower + Charisma (3) Test to make any quick decisions in situations they have not planned for.
Similar Archetypes: Authority, Schemer
I should note that I modify the base mentor spirits to give a little more freedom in terms of choice, hence why the above has a choice for all advantages.