Kerry-
Hello and Welcome to G-Nutz2!
Broad, open-ended questions garner broad, open-ended answers.
Any of us can give you our personal preferences as to parts, but those may not be what you ultimately like once you get them into the guitar.
And, I'm not an LP owner (yet), so I'm sure our LP contingent may be able to "zero in" on specific parts a bit better than I can. But several thoughts anyway . . .
First, you need to decide whether you want this to be as close to a "Gibson factory" LP as possible. If you want to be true to the original, then original parts are the way to go. One could, of course, just buy everything directly from Gibson, but that might get a bit pricey.
Second, since this sounds like a fine piece of wood you'll be massaging into your LP, you'll want to use high-quality parts to get a top-end result.
Third, an "LP" nowadays denotes a style of guitar moreso than a specific size of things. There are numerous variations worldwide. If your plans include a routing template, you'll want to be sure to find out whether the dimensions are for std. US-spec parts or not (from Stew-Mac, I'm guessing it would be so, but who knows?).
Now, since you are building from scratch, you're not wedded to factory specs; you can "build to the part", so to speak. This, of course, means having all parts in hand before you start drilling holes . . .
Fourth, one basic decision you need to make is whether you want to use the traditional Gibby "PAF" humbuckers or perhaps P90 pickups, which are also used on some LPs. Of course, you could use something else, too, but then it won't be quite as "LP-ish". I'm a fan of P90s on LPs, but let your ears be the judge on this. If you haven't heard both types, go to a guitar store and play each type of LP to see which sound appeals to you more.
Now, as far as a parts list, the Gibson website may have a complete LP parts list- I haven't checked, but many manufacturers include a parts list in their "technical help" sections.
You'll need tuners, of course, and quality matters here. Gibson currently uses Tone Pro tuners, which is now also Kluson since they bought the name. These would be the classic choice. I'm personally a fan of Grovers, they'd work fine as well. So would Sperzels. Be aware that the holes sizes for different brands will vary. "Measure twice", etc. and have tuners in hand before you drill the headstock.
Be prepared to spend a chunk of money for quality tuners. This is one area where it really does matter!
You'll need pickups, of course, and mounting rings. If your LP is to be a carved-top type, you need the pickup mounting rings for a carved-top body (the curved ones).
LPs come with or without a pickguard. If you want one, you'll also need the metal mounting arm (and screws).
The bridge is classically a "Tune-O-Matic", often called a "TOM", with a stop tailpiece. You can buy an actual TOM, or there are numerous clones available.
As far as electronics, and assuming this is going to be std. LP stuff, you'll need 4 500KΩ potentiometers. You will need the "long-shaft" types in order to go through the LP body.
You will need a 1/4" output jack and a jack mounting plate. LPs use an oval mounting plate, these are available all over the web.
You will need the two plastic cover plates to cover the rear control cavities. Often, one can find sets of "LP plastic parts" which include the back covers, pickguard, truss rod cover, and pickup mounting rings, and sometimes knobs as well. Also, these usually have the switch surround ring as well.
The quality issue doesn't matter much for the plastic bits, so look for a complete set if you can. This will at least ensure the colors all match up well.
You will need a 3-way pickup selector switch. This is a SPDT toggle switch. Make sure the plastic cover over the switch lever matches the rest of your plastic parts. A search for "Gibson style toggle switch" will disclose hundreds of these.
Additionally, you'll need a nut, wire (22-24 ga.), and strap buttons.
You didn't mention whether you were fabricating a neck as well, if so, you'll need fretwire and perhaps a truss rod and fingerboard. Neck construction is a whole 'nother subject which I'll leave for now . . .