A while back I created a sci-fi campaign using the Big Eyes Small Mouth system, that was pretty Shadowrun in feel (although it was before I played SR). I thought it would be interesting to ressurect the campaign using the SR system. The only thing that would really cause trouble is that my campaign had no magic. It did have psychic powers, but not to the extent that magic pervades in SR. I thought about just relabeling the system 'Mind Powers,' nixing spirits, projection (probably), and a few of the spells that would not mesh with being described as psychic powers, but at the same time I don't want to completly neuter anyone thinking about being a mage.

Does anyone have any ideas about doing a quick and dirty conversion of this type?

you could reduce the build point cost for magic (or the priority for the ABCDE system, make it cost a C or something instead of an A/B)

If you are going to keep adepts, then using the 'magicians path' adepts in Magic in the shadows as A priority magic would probably do. A sorcered with astral projection is probably worth more than priority B, but is not reeally worth priority A. A physical mage who can't summon is still probably priority A.

Things you'd have to think about:

Will there Adepts?
What about Foci?
Initiation?
Metamagic?
Will spell categories all stay, or will some go?
(Psychic Fashion spell?)

Offhand I'd say disallow full magicians, and no shamans. Magical or Phys adept paths are likely, reflavored under psychic power stuff. I remember a Batman Beyond episode I think where one of the badguys was essential a physad, but psychic background. Probably not so much in the way of summoning spirits. You may want to adopt some of the rules variations on making magic use in SR more frequent, allowing the psi's to essentially use their powers more often to make up for not having access to as many. Reduce the drain costs, etc. If using Anime as a background reference, unless you're talking Tetsuo level psi use, generally psi doesnt seem to be super debilitating. So it could be reflected as easier drain resistance.

I'm leaving in phy ads, because their style of powers in perfectly in line with what I had in mind. Since I was excluding spirits, that pretty much precluded any use of Shamans.
Intiations (karma expenditure, characters intiate themselves) and meta magics were left in so that characters could advance their powers.

Foci really don't fit in with this setting, so they are getting left out. Psionic powers are a recognized phenomena, but not as widespread as magic is in Shadowrun. Psionicly gifted humans are a rare occurence. As for spells (going by BBB):
Combat Spells:(cosmetic changes) the power etc spells are expressed in a pyrokinetic manner
Detection Spells: fine as is
Illusion: Not so sure with these.. They can't affect anything without a mind, so out with Improved Invisibilty, Physical Mask, and the like.
Manipulation: Good.
Elemental manipulation: Bad. It's gone.
Telekinitic manipulations: Very good.
Transformation Manipulation: Iffy. Petrify is defintely out. Light and Shadow too, because they don't really fit the setting.

Actually I would remove Combat spells (except Stun Bolt and StunBall) and add back in Elemental Manipulations as these are more easily explained as pyrokinesis, electrokinesis, cryokinesis and so on.

Manabolt makes a pretty good psychic heart stopper, or scanners like head burster.