A fun, catchy way to say the opposite of a 'no-brainer'?

Head-scratcher

2 Informal (originally US). A perplexing problem or question; something which causes bafflement or puzzlement.

https://en.oxforddictionaries.com/definition/us/head-scratcher

Although OED says that this term is originally AmE, it is certainly widely understood in BrE.

conundrum

Definition of conundrum, from Merriam-Webster. Only definition 2b, which I bolded, applies to your question, but conundrum is a fun word (at least IMO), and having three so different definitions adds piquancy to the word.

1a. riddle whose answer is or involves a pun (as in “Why didn't the lost hikers starve in the desert? Because of the sand which is there.”)

2a. a question or problem having only a conjectural answer •… the political conundrums involved, particularly the problem of how the richer areas … can be made to subsidize the poorer. — Douglass Cater

2b. an intricate and difficult problem •He is faced with the conundrum of trying to find a job without having experience.

If the OP insists on brain in the answer, this won't work.

Stumper

An extremely difficult question, task, or problem

Brain Buster (N) A particularly difficult problem or puzzle.

See also: "Brain teaser", "Head scratcher", "puzzler", "quandary"(N. A state of perplexity or uncertainty over what to do in a difficult situation. "X was in a quandary") and, of course, the other answers like "conundrum". (a favorite of mine)