Some common “spider” tropes to consider, based on various attributes of spiders; many of these can be played up and portrayed in a positive light.

Web Weaving

Probably the most noted characteristic of spiders is their (mostly) unique ability to spin webs. Spider silk is tremendously strong for its weight (to the point that modern attempts have been made to domesticate spiders for the purpose of harvesting the silk), and the webs that spiders produce are often beautiful.

Spiders as artisans

The most direct association with web-weaving is to that of craftsmanship. A spider god as artisan god is no great stretch, and the beauty, strength, and utility of spider webs makes for a solid demonstration of the association.

In real-world folklore, the story of Arachne is probably the most famous. In Ovid’s version of the tale, following typical Greek tropes, Arachne was punished for her hubris, but even the enraged and humiliated Athena left her able to continue her beautiful weaving. In another, Athena does win the contest, and by the rules Arachne could never again touch a loom or spindle again: there, Athena transforms her into a spider so she could continue weaving without a loom or spindle; a consolation prize.

Spiders associated with fate

Because weaving is associated with fate (specifically, the Fates of Greek mythology), and spiders with weaving, it is no great stretch to associate spiders with fate. I am not aware of any real-life folklore that makes such an association, but the opportunity is certainly ripe and should resonate with players aware of both “the tapestry of fate” and spiders’ weaving.

As noted below, spiders (and their venom) are associated with assassination, which is itself often associated with fate (cf. religious assassins acting as the hands of fate).

Spiders associated with stories

Weaving doesn’t have to be literal: we often speak of weaving tales and stories. The careful, crafty nature of a spider is also a good metaphor for the way a storyteller spins a story (and the choice of verb here is very much intentional).

From real-world folklore, Anansi is an African god of stories. He’s smart and clever, and is one of the most important gods in that tradition. Particularly in the Caribbean, he is associated with slave resistance and survival. Definitely a possibility for a CG god.

I’m not overly familiar with the real-world folklore surrounding Anansi (having only just read the Wikipedia page), but for an interesting portrayal of the character, Neil Gaiman’s American Gods and Anansi Boys deals with him (though he is not the central character of either). Good reads.

Spiders as mages

Like stories, spells are often described as woven. Spiders are also frequently associated with secrets and arcane knowledge, which obviously fit hand-in-hand with spellcasting.

For instance, spellweavers in D&D itself are eight-limbed (two legs, six arms) humanoids with considerable mastery of spellcraft. The association with spiders should be apparent.

Venom

All spider species, save for one family that lost its venom glands over the course of evolution (262 species out of appoximately 40,000), produce venom of some kind, and almost all actively use it in predation or self-defense. Most spiders are strictly predatory, though new research has revealed that plant matter is involved in the diets of a fair few species and access to plant-based food sources improves their longevity.

Spiders as healers

Spider venom can have medicinal properties, and spiders have been associated with potion brewing and the like. Furthermore, because of spiders’ associations with knowledge and secrets, a spider might very well be the place to go for the secret remedy to an ailment.

For a real-world context, I’m actually going to draw a parallel between spiders, and another famously-venomous animal: the snake. Snakes have long, in our world, been associated with medicine; one of the most well-known symbols of medicine is the Rod of Asclepius, a staff with a snake wrapped around it. Perhaps for this culture, in this world, that association fell to the spider rather than the snake: after all, only some snakes have venom, but (almost) every spider does.

Spiders as freedom fighters, defenders of the downtrodden

Venom is, in many ways, an equalizer: a way for a smaller, weaker target to defend itself from a larger, stronger foe. That is how many spiders use their venom, and it has historically been an aspect of poison for humans as well. Some historians have even posited that the negative association of poison, above and beyond other lethal weapons like swords, has as much to do with it being a threat against the establishment (who can protect themselves against more visible threats).

Thus, in a smaller, weaker race, spiders may be seen as mentors and guides, cunning tricksters who know how to best stronger foes. Anansi, mentioned above regarding storytelling, is this; in the Caribbean he is celebrated as a symbol of slave resistance. Mephala, mentioned below in the section on assassins, is also, the daedra who taught the dunmer how to protect themselves and eliminate threats.

Spiders as assassins

An extremely common trope, though usually one of the stand-out “evil spider” tropes, poison’s use in assassination and the well-known venom of several spider species makes for a very frequent pairing.

That said, positive portrayals of assassins exist in fiction. The concept of carefully eliminating one source of evil, killing one to save a thousand, etc., shows up from time to time. Definitely a controversial sort of good, and probably not enough to justify a Good label for a spider goddess in the eyes of players, it’s still worth considering, perhaps as a darker facet of a generally-wholesome goddess; even Good needs to go to war sometimes in D&D.

Furthermore, many somewhat-positive portrayals of assassins show them as being agents of fate. Again, due to the whole tapestry of fate thing, spiders fit into this motif exceedingly well.

For an example, I’m going to point to the Morag Tong of The Elder Scrolls III: Morrowind, where the traditional assassins’ guild is a cult to Mephala, a daedric “goddess” who is revered in both the traditional daedric cults of the dunmer and in the Tribunal religion that dominates the province in the time of the game. Daedra certainly aren’t good, but the dunmer generally viewed her positively – Mephala taught the dunmer how to defend themselves and destroy their enemies. During the time of the game, the Morag Tong was effectively the way in which battles between the Great Houses were fought, rather than open conflict; this was seen as saving many lives and much destruction. Compared to the Dark Brotherhood and Cammona Tong, the Morag Tong definitely has a more positive portrayal in the game.

Living in darkness

Spiders are one of the most immediate “creatures of darkness” in the human mind, along with bats. While certainly something played up quite a lot in negative portrayals and associations of spiders, it can have positive aspects too.

Spiders and secrets

Spiders are often associated with secrets and information-gathering. One does not need to look far for a spymaster called “The Spider” or similar. Varys in A Song of Ice and Fire comes to mind first for me, but he’s not alone.

Secrets are often dark, and those who trade in them often depicted as sleazy; Varys is certainly such an example. And yet no one would discount the importance of secrets, and to many cultures, particularly fictional ones, secrecy may be critical to survival. Certainly the Elder Scrolls dunmer, mentioned above, appreciated Mephala’s ability to keep them hidden.

Secrets also tie into both spellcasting and stories; Anansi, the spider god of stories above, is also god of secrets.

Preamble

Let me start by stating two points that are relevant to this answer. Feel free to skip this section if you want to jump directly to the 5e material.

1. In previous editions, official FR material dictated that paladins receive their powers from deities. In general, deities are a major part of this polytheistic setting, and mortals and powers are well-aware of this. Deities get to run the (meta)physics of the world and not having faith diminishes their power and causes imbalance. Our real-world secular/religious sensitivities are not attuned to this way of thinking, perhaps a good analogy is having your children vaccinated. When most people have vaccination, we get a herd immunity; likewise when most people have faith, the FR world 'runs'. As such not just paladins, but all people are expected to have a patron deity. If not, they are "disciplined" in the afterlife and become part of the "Wall of the Faithless". The attempts by the god of the dead, Kelemvor, to resolve this seemingly unfair treatment of the faithless has been depicted to have serious consequences in novels, which resulted in a change of his personality.

2. In spite of the central role faith plays in the FR of Greenwood, you are more than welcome to reject it; or find ways around it (like a faithless paladin who is still being watched over by a deity like Torm). Throughout its whole history, Ed Greenwood and the various versions of any written material have always highlighted the fact that each FR campaign belongs to the individual DMs. So feel free to rule however you like.

As for the officially printed material you have asked for, below is what is available.

What do the 5e Sources Say?

Sword Coast Adventurer's Guide states (page 131):

When ... a warrior also has great devotion to a particular deity, that god can reward the faithful with a measure of divine power, making that person a paladin.

And yet on page 132:

Most paladins in the Forgotten Realms, like clerics, are devoted to a particular deity.

These two sentences might seem to be in conflict, unless we interpret them to state that 5e FR paladins get support from one or more deities. This interpretation is supported by lore from earlier editions (see for example the Triadic Knights) as well as the text on clerics (page 125):

A typical cleric in Faerun serves a single divine patron, but some individuals feel called to serve a group, ...

D&D Adventurer's League

It is worth noting that the D&D Adventurer's League FAQ has switched its position on this issue over time. (The change might be correlated with the introduction of non-FR campaigns to DDAL.) In version 2.5 it stated:

Does My Paladin Have to Worship a Deity? Yes, though your character’s alignment isn’t required to match that of their deity.

While version 3 states:

No. What makes you think you do? Dunno what you’re talking about.

WotC Suggestions for Content Creators

Finally, Wizards of the Coast has a set of style guides for their official play writers and have been made available for all content creators on DM's Guild. Note that these guides are suggestions and not rules, but for the sake of completeness, I would like to quote the relevant parts. D&D IP Guide states:

Divine magic is practiced by the most devoted followers of D&D’s gods: clerics and paladins. ... Paladins are the holy warriors of D&D. Like clerics, they serve a deity and channel divine power to achieve their goals (or the goals of the temple or holy order they serve).

Forgotten Realms Style Guide confirms:

Paladins are warriors of unusual virtue and piety rewarded by a good god with divine power. ... paladins are typically judged more swiftly and harshly by their gods.