In Skyrim, what variables can increase the duration of invisibility spells?

Dual-casting invisibility with the relevant dual-casting perk will increase its duration to 66 seconds. This is not magicka-efficient compared to re-casting the spell right before it runs out with the silent casting perk, and it's still rather short.

"Fortify [spell school]" effects come in two varieties (which the active effects screen distinguishes by using the words "fortify" and "magnify" respectively):

• Make [school] spells cost less magicka - Granted primarily by enchantments (whether naturally-occurring or artificially crafted). With the right combination of gear, they can reduce the casting costs of one or two schools (with dual enchanting) to zero. However, these are useless for OP's purposes.
• Make [school] spells more powerful - Granted primarily by potions (whether naturally-occurring or artificially crafted), as well as the Blessing of Mara (Restoration only). The definition of "more powerful" varies by school. For Alteration and Conjuration, this actually does increase the duration of the spell, but for Illusion, Destruction, and Restoration, it increases the magnitude instead. For Illusion, the magnitude is the maximum level of creatures affected by the spell, while for Destruction and Restoration, it is the amount of health restored or destroyed.

Because the Magnify Illusion effect does not increase duration, there is no general-purpose means of making Illusion spells last longer, other than dual-casting them.

The best way to do so is to install a mod.

The Invested Magic mod makes most of those type spells sustained. While active, they remove a portion of your maximum mana cap instead.

In the base game, there is no way to lengthen the duration of an invisibility spell but there are invisibility potions that last longer than 30 seconds.