"Back up data" or "back data up"?

Solution 1:

Back up data is by far the most common, (referencing two Google searches (1780000 results vs. 18200) and Ngrams), and sounds the most natural. This is because back up, though written as two words in verb form, is spoken (and thought of) as one word. To split the "verb" up and put something in between (e.g. data) would be to confuse the meaning of the sentence for most people.

He was careful enough to perform tests and back up data to avoid any problems.

Backup as one word is a noun/adjective, and is not applicable as a verb.

The backup software on this computer is ancient.

The backup runs from 12 AM to 1 AM every night.

Edited to say that I wholeheartedly agree with @tdhsmith's comment:

...while keeping the "verb" whole is generally more readable with wordier objects, most native speakers must split it when using a pronoun. For example, "I back it up every night" is correct to me but "I back up it every night" feels very wrong. [See below for the entire comment]

Solution 2:

Normally, back up as a verb is seen as a compound form that shouldn't be broken, so OP's first example is the standard phrasing. But all credit to @lettuce_pants for pointing out that you can (must, in fact) split the words when using a pronoun - for example, "I back it up every night".

That compound (technically, a phrasal verb) is often collapsed into the single word backup (almost always, when used as a noun/adjective as in backup data, backup plans, etc.).

OP's second example is far from unknown - quotated "back your data up" gets over 500,000 hits on Google. But there are 2.5M for "back up your data", and 1.8M for "backup your data".

So whilst I personally don't think it's meaningful to talk about "correct" usage here, the safest, easiest option is to always use backup. If you're confident you always know when you're using it as a verb, insert the space (but no other words apart from pronouns) in that context. Then no-one will ever be able to find fault, no matter how pedantic they are.

Solution 3:

"back up data" is probably better, if you consider back up to be one term/word, ie. "back up", "back-up", "backup".

The only problem is that "backup data" could refer to the data itself rather than the operation. eg. "The backup data is on tape" rather than "we backup data onto tape"