Can I do anything with all of my extra SP?

As of the most recent update to the game (Ver. 1.2.0), the extra SP accumulated by heroes can now be used after having that hero Inherit Skills from another hero.

From the in-game FAQ on how Inheriting Skills works:

A Hero can inherit learned or unlearned skills from another Hero, but it's important to note that a Hero does not immediately learn a skill just by inheriting it. In order to learn the inherited skill, the Hero will need 1.5x the normal amount of SP required to learn that skill.

The 1.5x multiplier on skill cost will likely burn through a lot of the stockpiled SP of maxed out heroes.

As of version 2.0.0, units can improve their weapon in the Weapon Refinery, though not all weapons can be refined.

Aside from Arena Medals, Refining Stones, and Divine Dew, refining weapons also costs SP.

Weapons that require Refining Stones cost 350 SP to refine. For those that require Divine Dew, the cost is 400 SP.

Some weapons can even transform into a new weapon, which can be refined even further. The most common example are the Killer/Killing weapons, which evolve into Slaying ones.

(Answer was accurate pre-v1.2.0, refer to Vemonus's answer for more recent information)

At the moment, it doesn't look like it. Even if you were to merge that character with another (not that you would with a maxed 5*), it wouldn't even give any more SP to the recipient. Maybe somewhere down the road they'll add something to do with SP after maxing out a hero, but for now it just builds up.