The definition of Chess.com's original computer evaluation of "brilliant" moves was never revealed but people speculated that it used to identify moves as brilliant where the computer evaluation changed positively once a higher search depth was reached. (Because it was seeing something in deeper analysis that would mirror the intuitive grasp of a human played brilliant move.)

The problem was the moves identified this way often didn't appear "brilliant" to humans. 

Chess.com's staff worked hard on improving the definition several years ago and decided that a key element of a ''brilliant" move was a sacrifice.

And they also decided that while the benefits of certain sacrifices might be obvious to higher rated players, they were far from obvious for lower rated players and might require an element of a brilliant move.-- so they would call moves "brilliant" for lower rated players that were obvious for higher rated players.

/ from the Help page:

How are moves classified? What is a ‘Blunder’ or ‘Brilliant’ and etc?

After each game, you'll see a list of all your moves in the game review, classified as 'Best' or 'Inaccuracy', or many other such classifications. How are these determined?

Move Classifications

Classifying moves is a mix of art and science. Where is the line between a good move and an inaccurate one? How is a blunder defined for a chess master compared with a new player? What matters more, going from +2 to +1 or from +0.7 to +0? What engine evaluation is needed for a position to be considered “winning”?

With ClassificationV2, Chess.com has moved to an “expected points” model, rather than strict evaluation differences, to answer these questions. 

Expected points uses data science to determine a player’s winning chances based on their rating and the engine evaluation, where 1.00 is always winning, 0.00 is always losing, and 0.50 is even.

Basically, at 1.00 you have a 100% chance of winning, and at 0.00 you have a 0% chance of winning. After you make a move, we look at how your expected points (likely game outcome) have changed and classify the move accordingly. The table below shows the expected points cutoffs for various move classifications.

Table I: Move Classifications with their corresponding change in expected points boundaries. If the expected points lost by a move is between a set of upper and lower limits, then the corresponding classification is used.

Special move classifications that use rules beyond expected points have also undergone improvements. This includes familiar classifications like Missed Win and Brilliant, as well as the new Great Move classification.

A Missed Win is when you miss an opportunity to capitalize on your opponent’s mistake and gain a winning position, and instead end up equal or worse. As with expected points, the engine evaluation needed to be in a winning, equal, or losing position will change along with a player’s rating.

Brilliant (!!) moves and Great Moves are always the best or nearly best move in the position, but are also special in some way. We replaced the old Brilliant algorithm with a simpler definition: a Brilliant move is when you find a good piece sacrifice. There are some other conditions, like you should not be in a bad position after a Brilliant move and you should not be completely winning even if you had not found the move. Also, we are more generous in defining a piece sacrifice for newer players, compared with those who are higher rated. 

Great Move is a new move classification that is denoted by a single exclam (!). These are moves that were critical to the outcome of the game, such as going from losing to equal, equal to winning, or finding the only good move in a position. Similar to Brilliant moves, we are more generous on what we call a Great Move for new players compared with high-rated players.

Overall, the improved Move Classification system provides a tailored Game Review experience. The new definition of Brilliant and the Great Move category give you a way to identify and share the most interesting moments from your games. The expected points formula identifies the mistakes that matter most for your improvement.