Learn Finnish: Voinko sanoa sanasen? - Can I say a quick word?

The suffix -nen acts as a diminutive here, turning sana (word) into sananen (a little word). In this context, it softens the request, making it sound more polite, humble, and less demanding, much like saying "a quick word" in English.

Why is the word sanasen and not sananen?

The verb sanoa needs an object. Since you are asking to say one complete thing, it takes the total object case (the genitive-accusative), which ends in -n. Words ending in -nen change their stem to -se- before this ending, so the nominative sananen becomes sanasen.

If I wanted to make the sentence negative, how would the object sanasen change?

In a negative sentence, the object must be in the partitive case. The partitive form of sananen is sanasta. So you would say: En voi sanoa sanasta. This is a great trap for learners because sanasta is exactly the same as the elative form ("from/about a word") of the base word sana, but the context here makes it clear!

Could I just ask Voinko sanoa sanan?

Yes, absolutely. Voinko sanoa sanan? is grammatically correct and simply means "Can I say a word?". Choosing to use the diminutive form sanasen instead is just a stylistic choice to make your interruption feel smaller and more courteous.