Now that Friends in High Places has been officially removed from play (MWL 2.0), this card makes a lot more sense.
To consider the value of corp recursion, it helps to compare it to its core counterpart, Archived Memories, which I think set a pretty good precedent for getting stuff out of archives.
Archived Memories - For a click and a card with two influence, you get to add a trashed card directly back to HQ.
Let's compare that to one of Jackson's post-rotation replacements, Preemptive Action:
Preemptive Action - As your last action of the turn, you may spend a click to recur three cards from archives, but must tutor or draw them again naturally to get them back into HQ.
Overall, similar prices with slightly different uses. Outside of HB, Preemptive Action is probably an easier choice.
FiHP - As your last action of the turn, you may spend a click, two credits, and a card with one influence to recur two cards from archives and install them.
To achieve the same thing with Archived Memories you'd have to have two copies (potentially costing 4 times the influence), play both, then spend two more clicks installing them. The cost for saving three clicks, three influence, and a card slot? Two credits.
But let's not dwell on that.
Restore - For a click, a credit (plus the installed card's rez cost), and a card with three influence, you may recur a single card, install it, and you must rez it.
To achieve the same thing with Archived Memories, you'd have to spend a click and a card, and then another click to install the new card (rezzing it whenever you need to).
Effectively, assuming you intend to rez the installed card, you are spending a credit (and maybe an extra influence) to compress two clicks into one. (Also, the more copies of said card in archives, the less often you can do this)
In general, that's not a good deal, but if you really need assets, upgrades, or ice back fast, this will do the job. It's exactly what FiHP should have been all along.