Short Answer: The word πειθω has the basic sense (in the active voice) of "convince" or "persuade" (cf. Gingrich) and καρδια just means "heart". So literally it would be "convince our heart". However, the reason there is so much debate surrounding the translation of this phrase is not that the terms are unclear, but that there is a bigger theological debate going on in the background which is driving the translation choices. "Reassure our heart" has become associated with the view that John is encouraging his readers, while "convince our conscience" has come to be associated with the view that John is confronting his readers and driving them toward repentance.
This is an interesting discussion which actually ties in with several other interpretive challenges in the surrounding context. Perhaps the easiest way to explain it would be to survey the two competing interpretations.
View #1: John is reassuring his readers
In this view, John wrote his letter in order to reassure his readers of their salvation, which was evident by their love for one another. In the more immediate context, John has encouraged them to continue loving one another (3:18), by which they will continue to know that they are right with God (19a; notice that the NA28 ends the sentence mid-way through verse 19.)
Furthermore, by continuing to love one another, if our heart happens to try to condemn us, we will be able to reassure our heart that it is wrong because God is greater than our misleading heart, and He knows everything, and He has said that those who love one another are right with Him (19b-20).
The goal is to get to a place where we don't feel unnecessarily condemned, because then we will feel comfortable presenting our requests before God (the basic meaning of the word translated "confidence"), and so we will present our requests, and we will receive what we request, because we are doing what is pleasing to Him (21-22).
For what it's worth, I think this is the plain and obvious meaning of John's words here, though I will grant that the syntax is challenging in this section.
View #2: John is confronting his readers
In this view, John has just said something very challenging: that if you hate your brother you are a murderer (3:15), that we are obligated instead to lay down our lives for our brothers (16), and so closing our heart (lit: stomach) to our needy brother is not loving (17). We should not love with our words alone but with our actions (18; notice that the UBS4 ends the sentence here instead.)
Now, the logic goes, who can read such a statement and not feel convicted? This surely would have served to confront the readers in many of the areas where they were unloving -- and that's good; that's exactly what John is trying to do. John wants them to understand what love looks like so they can see their shortcomings and repent.
By this new found understanding we will be able to come to know that we are from the truth and we will convince our conscience in God's presence (i.e. to repent and do the right thing by loving one another) whenever our conscience rightfully condemns us, because God is even more severe in His judgments than our conscience; He knows everything.
If our conscience is not rightfully condemning us about anything, on the other hand, then God will hear our prayers.
Summary
As you can see, though the Greek literally just says something like "convince our heart", there are two very different ways of interpreting the meaning of this phrase in context. This has led to two very different modern interpretations, with the first being the idea of reassuring our heart when it wrongfully condemns us, and the second being the idea of convincing our conscience to lead us in the right direction based on the fear of God and knowledge of His ways.
Personally I think the second view is outlandish and the first is obviously true, but my preference is to simply translate it "convince (or persuade) our heart" and then explain the sense of the phrase in context as needed. I think most people out there would not naturally take John's statement as a confrontation anyway.