I might be able to shed some light here and there for a few different pre-Columbian cultures.

The most intimate understanding of eclipses and their cycles in the New World came from the Maya. The Dresden Codex (or was it the Paris Codex? I'll see if I can check it, and edit here), one of only five remaining Maya codices, details at one point an eclipse table - a count of the days intervening between eclipses. These days between eclipses follow a pattern, centered around 173 days, which is a half-year minus the regression of the lunar nodes (as the Moon's orbit if five degrees off the ecliptic, there are only two points where the Sun-Earth-Moon system is truly lined right up, and these two points regress with respect to the ecliptic). Basically the days noted were 173, plus or minus a month, or plus or minus two weeks - roughly. This table and the pattern it notes implies that the Postclassic Maya at least had abilities to anticipate lunar and solar eclipses. (Recent discoveries by William Saturno of intact murals at a Classic site also have these eclipse tables - so this knowledge may have been in use for some time.)

So, what to do with a table of eclipses? To the Maya, the next step was planning your party. Many accession dates on Maya stelae - dates noted by city-states of the crowning of their next ajaw, or ruler - sync up with eclipse dates, as well as conjunctions of planets. The Maya also had excellent observational understandings of the cycles of Venus (Kukulkan to them, later Quetzalcoatl to the Mexica), Jupiter, and Mars. Predicting and timing large festivals to coincide with planetary conjunctions and/or eclipses lent an air of supernatural legitimacy to these rulers, claiming the throne with the power to change the heavens. To the Maya, knowing these cycles was a good sign, and would have been well-anticipated.

Stories of the Inca in the Andes and Aztec Triple Alliance are somewhat less flattering with regard to their knowledge. The Inca supposedly knew little of how eclipses worked - solar eclipses of the sun, Inti, were quite worrisome, with common folk throwing stones at the sky in an effort to aid the sun to regain the light. If I remember correctly (and I know my Mexican-flaired brethren can probably describe it in more detail), central Mexican groups of the Postclassic saw a solar eclipse as the Sun being eaten by a demonic disembodied head - he would swallow the sun for a moment before it slipped through his open throat and back into normalcy.

I'll see if I have any more resources about eclipses among the Inca - lunar eclipses especially. Fascinating stuff.

To understand what's going on for the Haudenosaunee (Iroquois), we have to take a few steps back in the Creation mythos.

Sky Woman fell to earth while pregnant and gave birth to daughter named Lynx, who in turn died while giving birth to twin boys. After her death, Lynx became the spirit of the earth. Sky Woman raised her grandsons until her own death, at which point she became Grandmother Moon and continued to watch over her great-grandchildren, humanity. In some versions of the Creation, Sky Woman and Lynx are merged into one character, but since Iroquoian metaphysics gives people two 'souls' (one that travels skyward after death and one that wanders the earth), Sky Woman can be both Mother Earth and Grandmother Moon without any problem.

When the moon reddened or darkened during an eclipse, it meant that Grandmother Moon was either displeased or ill. She needed to be appeased or healed. Unfortunately, my source doesn't give any examples of how Grandmother Moon was appeased or healed. Dances seems to be the most likely method, as the spirits love to see people dancing and dances were often used as community activities in association with healing. Dances could also be organized and carried out within the span of an eclipse.

Sources: The League of the Iroquois, Lewis Henry Morgan

Iroquoian Women: The Gantowisas, Barbara Alice Mann