Learn Russian: Секундочку, я ищу самый важный документ. - Just a second, I am looking for the most important document.

Секундочка uses the diminutive suffix -очк-. While diminutives are often taught as meaning something is physically small, here it is used pragmatically to soften the phrase. Saying секунду (a second) can sound a bit blunt or demanding. Adding the diminutive makes the request sound friendlier, more polite, and implies the interruption will be minor.

Why is it секундочку with a -у at the end instead of секундочка?

It is in the accusative case. This is a conversational fragment that implies a dropped verb, like "wait for a second" (подождите секундочку). In Russian, expressions of time duration—how long an action takes—always take the accusative case.

Ищу looks very different from its infinitive. What verb is this?

The infinitive is искать (to look for). It belongs to a group of verbs that undergo a consonant mutation when conjugated in the present tense. The stem cluster -ск- changes to -щ-, giving us я ищу. The rest of the present tense forms follow this mutation (e.g., ты ищешь, он ищет).

Самый важный документ is the direct object of the verb. Why doesn't its ending change to show the accusative case?