Q:

What is the case of "einem" in this context?

Man hat so Dinge im Kopf, die einem wichtig vorkommen. Aber was im
  Endeffekt das Wichtigste ist, finde ich interessant zu sehen.

I can't analyze the case of "einem wichtig" in the above sentence. I take die wichtig vorkommen as a predicative sentence in the following form if we want to say in English:

They appear to be important.

And in German:

Sie vorkommen wichtig.

But then I cannot understand why "wichtig" should get "einem". 
So I then thought maybe "einem" is not an article for "wichtig" but means something like "at first look" or alike, but couldn't find anything about it. 

A:

The einem could be translated into English as to you (or a bit longer but closer to the original meaning: to the person making the observation).
So

Man hat so Dinge im Kopf, die einem wichtig vorkommen.

translates to

You have items in your mind that appear to be important to you

or a bit further away from literal translation but less ambiguous

You have items in your mind that you consider important.

A:

The Wiktionary entry for vorkommen lists characteristic word combinations. If vorkommen is used with the meaning [4] auf jemanden in einer Weise wirken, it is often used as:

[4] jemandem kommt etwas vor wie/als

where jemandem is in dative case.
Your example sentence follows this structure: einem is not used as an article, but as a pronoun (in dative case) that replaces jemandem, see this Duden entry for more examples.

