For God so loved the world that he gave his only begotten Son that whosoever believeth in him should not perish but have everlasting life.
There are a couple issues, of course, that lend themselves to misunderstand what this verse is actually saying. One of them is the term "world" in that it is interpreted to mean every single person in the world, rather than what John likely means in context, i.e., "not only people who are Jews, but people from all nations as well" (cf. 1:11-13).
But the larger misunderstanding comes from both the English interpretation "whosoever" and a misquote of the English translation "whosoever may" that one often hears in these circles. First, there is nothing here that says "whosoever may" and even if one were to translate it this way from the subjunctive of ἀπόλλυμι, it doesn’t convey that anyone “can” believe, but that what God had done in giving His Son was so that they “may not perish,” the “may” or “might” conveying the intent of God’s giving, not the ability of the humans to believe.
The word πιστεύων is a present active participle, not a subjunctive verb; but even if it was, this does not convey ability to do the action, but the possibility of the action being performed. In other words, X might do Y, but this says nothing as to why X does Y, or whether X had the ability on its own to accomplish Y. In any case, the idea that “may,” if one were so inclined to translate it this way, conveys “ability to do so” is not what the Greek conveys.
Which leads us to the next point. The participial construction ὁ πιστεύων doesn’t mean “whosoever.” This is the original translation of the Geneva Bible, and it was picked up by the KJV (I'm unaware if it goes back earlier than this), and it has thus become our common understanding of the passage, i.e., our tradition. Now, I have to say, there's nothing wrong with translating it this way. The Geneva Bible itself was a Calvinistic translation and had no problem with it; but it wasn't horribly misused in its day either. So it can be used and understood in a Reformed setting. It just needs to be understood in light of the original language.
So what does it mean? The participle is likely meant to be a substantive, "the believing one," or "the believer." So how did we get "whosoever" out of that? Well, here is what I think happened. In English, we often translate substantival participles as relative clauses. This is not the intent of the Greek, but the Greek doesn't come over well in the flow of the English language at this point. Hence, since it sounds weird to translate it as "in order that the believing one in Him might not perish, but have eternal life," we translate it "he who believes." But in light of the context, which includes Gentiles as well as Jews, the translators seem to have felt it better to expand upon the word "who," i.e., the word that doesn't actually appear in the Greek, and translate it as "whosoever." Now we end up with an expansion on a word that most people emphasize when they read the passage--again, a word that isn't there. Relatives are conveyed with relative particles. No such relative particle appears here. Unfortunately, many a Greek student will be taught to translate, and even attribute a relative meaning to a participle because this is what "sounds better" in English, but this is merely a gloss, not what the original language is conveying.
So what does John 3:16 actually say? It simply says that "God loved the world so much that He sent His unique Son in order that the believing one in Him would not perish but have eternal life." In other words, God sent His Son for the one who believes (see, I can't even say it English without a relative clause). The question as to why he believes is not answered here, and I would say the same thing to those who retain the reading, "whosoever." The passage only explains that God sent His Son to save those who believe. Why they believe is explained more throughout the rest of the Gospel (e.g., in 6:36-47).
So, no, it doesn't say that anyone has the ability to believe. In fact, those who use it this way set up a contradiction between what Jesus says here and what He says in the rest of the Gospel (see, for instance, 12:37-41). There is no word "may" here. There is no word "whosoever" here. There is only the expressed love of God in the giving of His Son to save the one who believes. Do we know who these believers will be? No, and that's why we can continue to use the word "whosoever," because whosoever you are, if you hear His gospel today, you are being told to believe, and if you believe, you know that God has loved you and has given you His Son for your everlasting preservation and joyful fellowship with Him. So, whosoever hears and believes, let him come.