What does "it doesn’t make much of a case for itself" mean?

The phrase is definitely correct idiomatic usage.

To make a case for something, when something is a noun (or pronoun), is to provide good reasons why the named thing should be considered for use at whatever tasks that thing is normally used for.

You could consider it to be a shortened form of make a case for you to purchase something.

If the device in question "doesn't make much of a case for itself", then it is failing to provide any compelling reasons for you to purchase and use it instead of one of the available alternatives; we can presume that its competitors are more capable, or cheaper, or easier to use, or better-looking, or some combination of those.

Edit to add: In particular, the "for itself" is appropriate because a salesperson could "make a case for the phone" by explaining its benefits and features to you. In the absence of a salesperson, the phone has to have enough features and benefits that you can discover without assistance: the phone has to sell itself, or "make a case for itself".

The phrase is fine, although I'm not a fan of hi-tech toys and the various statistics in the article leave my head spinning, the author of the review is comprehensible. Andrew Leach is correct by pointing you to read carefully the definition of: make a case for. What the dictionary says, is what is meant in the article.

• based on the legal meaning of make a case (to show that what you say is true)

It is as though the new smart phone/tablet (whatever) has to justify its existence in the ever growing, saturated, market of smart phones. It is the defendant defending itself, and in order to convince the jury (the consumers) to let him free (figuratively speaking) he puts forward his case.

According to the reviewer the defence is weak; doesn't make much of a case, thereby claiming there are better candidates which are more deserving to remain in the market.