Synthesis: combination
Topical preaching: "today the purpose of my sermon is to assure you of your salvation"
Exegetical preaching: "today we're covering 1 John 5:1-13"
0) Context
There is a reasonable debate between topical and exegetical preaching. One of my past churches recommends that you only attend a "bible-beleiving, exegetical church". A pastor of a church that I attended at the same time commented once that the debate between the two was silly, and proceeded to do topical preaching.
1) Purposes
a) The purpose of preaching topically is to help people in a certain area of their lives, to take care of a certain issue
b) The purpose of preaching exegetically is to hear directly what God has to say, not what we humans want him to say. It is also to make sure that we cover all of what God has to say (by going linearly through a book) rather than just the parts we want to use.
c) These purposes are NOT opposities
2) Synthesis:
a) Topical>exetegical: Start a sermon by pursuing a given truth from God. For instance, "What does God have to say about assurance of salvation?". Then, go looking for passages that discuss that TOPIC, NOT what you want to say about it. Do not go looking for a passage that says 'you know you are saved because god loves everyone' because you think that's part of the answer. Look for anything regarding assurance of salvation, dig the answer out of that text, and accept what you find.
b) Exegetical>topical: Pick a passage of the bible and look to see the various topics that it covers. Pick one, and make that the point of your message, filtering the passage for that specific lesson, emphasizing certain parts and not others. Each part of the bible has MANY lessons to learn, this ought to be easy.
c) Provided that you are taking whatever God tells you over your own ideas by searching the bible with that motive, and you are also seeking to apply whatever God tells you to real, heart, human issues, you should be fine.
3) Suggestions from a simpleton:
a) The first synthesized method has the advantage of addressing issues that you chose, so you can chose the ones you think are currently plauging that church or audience. This is very real, very useful and has great motivational/change making potential.
b) The second sythesized method has a wider range of good academic advantages. You can cover all of what God wants to say, probably in an order that he wants (because the bible is not arranged in some haphazard order), and you get bonus points for being more official. To help with the application-deficient side, by the way, you can choose a book or chapter using holy spirit methods (or just pick) and trust God to make sure that its' the right message at the right time, even if you can't see it.
c) I'm not sure which method is best (or if one is better than the other). I would suggest using the first in specific situations, like camps, holidays, sending off missionaries, etc, and using the second for mass congregation preaching. I DO know that all preaching I've heard to far could benefit from a greater trust and respect for the living word of God, and more real, challending ways to apply it.