Both O'Bannon and Geiger were big Lovecraft fans and said that Yog-Sothery was a major inspiration for Alien (incomprehensible alien monsters, uncaring universe, even details like the eggs being kind of Elder-Thing shaped).
How would you play that element up? What parts of the mythos would be a good fit?
Black goo has something of a Shoggoth (or FSoT) vibe to it (i.e. living black liquid, associated with ancient aliens, who suffered a disaster, possibly due to their creation running out of control). AI / synthetics hint at possibly 'going shoggoth' on humans a few times (most notably, David in Covenant).
Space jockey has a superficial resemblance to Rhan-Tegoth, Chaugnar Faugn, and Yag-Kosha due to the trunk (and being a huge, weird space alien).
Novelisation of Alien mentions some kind of futuristic trained dreaming. Think this is expanded a bit elsewhere with dream recordings or something. Could tie in to dreamlands, I guess?
A:E has transmission of consciousness (or at least a facsimile of personality) into new bodies. Big theme in Lovecraftian horror (Thing on the Doorstep, Shadow Out of Time).