What is the Book of Mormon equivalent of John 3:16?

I'm going to mention other groups of religions and there are, of course, variations within the groups, and it is not to disparage them, but to point out some of the differences and similarities. I have attended several of them and like the work they are doing in "all of Christendom" and in the world. Any disagreement part you will be able to pick up on.

Eric's answer is good because the two scriptures he mentioned show some of the similarities and differences between Mormons and Catholics and Protestants, right off. [James 1:5] Mormons believe that one can have a personal relationship with God as many of our Evangelical Protestant friends do. We also believe the Holy Ghost frequently and Jesus Christ and God the Father occasionally communicate with God-seeking humans as many of our Charismatic Evangelical Protestant friends do. Unlike our Charismatic friends, however, Mormons don't tell the Holy Ghost "come" in a controlling or commanding way, but rather ask and the Holy Ghost answers when He wishes or not at the moment. (Mormons do not believe in the Nicene-type Trinity Creeds of 325 AD and earlier, which most Catholics, Orthodox, and Protestants still hang onto even though there were many dissenters to the idea at the time. Mormons might say that the Baptism of the Savior show the 3 separate members of the unified Godhead.) [3 Nephi 11 onwards] Mormons believe that Jesus Christ visited, at least, 3 groups of people, Jews in Jerusalem, other Israelites and Native Americans they intermarried with in a small part of the Americas, and another group whose location was unspecified. It is likely that Christ visited others, that He chose not make know to the public at the time. The Hopi have a stone carving showing Christ's visit to them. The message of both scriptures, in a nutshell, is everyone can have a relationship with the Savior and "Seek Him, and you shall find Him".

One small, often quoted, scripture that also contains a concentrated dose of doctrine is:

Moroni 10:3-5

3 Behold, I would exhort you that when ye shall read these things, if it be wisdom in God that ye should read them, that ye would remember how merciful the Lord hath been unto the children of men, from the creation of Adam even down unto the time that ye shall receive these things, and ponder it in your hearts.

4 And when ye shall receive these things, I would exhort you that ye would ask God, the Eternal Father, in the name of Christ, if these things are not true; and if ye shall ask with a sincere heart, with real intent, having faith in Christ, he will manifest the truth of it unto you, by the power of the Holy Ghost.

5 And by the power of the Holy Ghost ye may know the truth of all things.

This scripture, located near the end of the book, emphasizes the nature of the Godhead and encourages seeking a direct and personal revelation as to the truthfulness of the Book of Mormon; as well as other things that you may wish to know the truthfulness of.

From the very title page:

... And also to the convincing of the Jew and Gentile that Jesus is the Christ, the Eternal God, manifesting himself unto all nations— ...

...to the last page of the record (Moroni 10:32-33):

32 Yea, come unto Christ, and be perfected in him, and deny yourselves of all ungodliness; and if ye shall deny yourselves of all ungodliness, and love God with all your might, mind and strength, then is his grace sufficient for you, that by his grace ye may be perfect in Christ; and if by the grace of God ye are perfect in Christ, ye can in nowise deny the power of God.

33 And again, if ye by the grace of God are perfect in Christ, and deny not his power, then are ye sanctified in Christ by the grace of God, through the shedding of the blood of Christ, which is in the covenant of the Father unto the remission of your sins, that ye become holy, without spot.

... the Book of Mormon espouses the doctrine of Christ: His gospel, ministry, and everlasting Atonement:

For another instance, 2 Nephi 33:10:

10 And now, my beloved brethren, and also Jew, and all ye ends of the earth, hearken unto these words and believe in Christ; and if ye believe not in these words believe in Christ. And if ye shall believe in Christ ye will believe in these words, for they are the words of Christ, and he hath given them unto me; and they teach all men that they should do good.

However, the overall focal point would be in 3 Nephi, chapters 11-28, which records the Savior's ministry upon the American continent. I have seldom seen Book of Mormon passages referenced in an introductory setting more than those which directly speak of Jesus' gospel: faith, repentance, etc.

The Book of Mormon repeats the same doctrine over and over but in different ways... so it's hard to pinpoint a specific verse to answer your question. Typically, missionaries will probably invite you to read in 3 Nephi about Jesus' visit to the Americas, or in 2 Nephi 31 where Nephi preaches on faith, repentance, baptism, and remission of sins.

Which passage missionaries assign people to read is chosen on an individual basis. To explain Christianity in a nutshell, they would probably refer you to John 3:16 in the Bible (it is an excellent verse) -- to explain Mormonism in a nutshell, it varies between any of these types of passages, depending on who you are and your situation.

Two ways to determine prominent LDS scriptures are the "scripture mastery" lists and citation data from General Conference sermons.

Scripture Mastery

The LDS seminary program encourages youth to increase familiarity with the scriptures by memorizing select passages. This "scripture mastery" program specifies 25 passages in each of the Old Testament, New Testament, Book of Mormon, and Doctrine and Covenants. Not that this is somehow a canonical list of the most important verses in scripture, but I suspect the list was chosen based on doctrinal relevance, frequency of use in sermons, etc. So, the Book of Mormon Scripture Mastery list is a good place to start.

References in General Conference

One practical measure of prominence is the frequency with which scriptures are quoted in General Conference. The Passages of Scripture Most Cited in General Conference, 1942 – 2004 was compiled using citation data from http://scriptures.byu.edu/. This list includes passages from not limited to the Book of Mormon.