What is the meaning of not in "as often as not" and "as likely as not"?

You are confusing two different constructions.

For the meaning you are after, you would have to use an actual quantifier: "as often as ten times a week", "as often as twice a year", "as often as every Tuesday", "as often as never". This follows the pattern of phrases like "as recently as last Tuesday", or "as soon as tomorrow". Basically with phrases of this kind you say that it's often/recent/soon, and then immediately quantify just how often/recent/soon it is.

Not is a completely different part of speech, it does not fit the pattern. "It happens as often as not" is a simple ellipsis of "it happens as often as [it does] not". So 50% of the time. This is a different pattern, compare it to another common phrase "more often than not". Which, again, does not mean "more often than never", but rather "for every time X does not happen, it does happen at least once". So 50+%. It happens more often than [it does] not.

Likewise for "as likely as not". If you are after solid examples, you can search the Corpus of Contemporary American English (COCA), which has 18 cites, or the British National Corpus (BNC), which has 21. Here are some of them:

• Of course, as likely as not, I'll throw it away.

• Me, I think it's as likely as not that I'll be on that particular plane.

• The wooden box to which all these fittings were fixed probably contained the gaming pieces, which were as likely as not also made of ivory.

• But last year, after a two-year battle, Devereaux convinced the Veterans Administration that his cancer was as likely as not linked to the water at Lejeune — something only one other vet had managed.

• The farm worker is as likely as not to be found standing shoulder to shoulder with his employer defending the farm and its game — a literal case of poacher turned gamekeeper.

• Then, it seemed hours later, when he sensed or smelt or somehow divined that he was almost at the road, there came, as likely as not out of his own imagination, the delicate sound of an indrawn breath.

• She would smile, and hand me the ball, which I would toss behind my back and catch in front of me, then roll down my arm and catch again (as likely as not dropping it and comically flailing into a routine of awkwardness and dismay) and pass back to her.

Follow the links for more examples.

This is the same ellipsis as in, say, as good as gold:

It is as good as gold (is good).

Everything except the contrasting material is deleted:

This is true as often as (it is) not (true).

If something is true as often as not it is true at least half the time, and generally more often.