Just a basic question since I haven't really figured it out yet.

If you give your companion NPC a new 'upgraded' weapon to use (stronger then their own) what determines if they use it or not? Do they have their own specialties? For instance, I have Lydia with my main PC and she's pretty set on keeping the non magical glass warhammer in use instead of a one handed weapon much stronger. She'll almost alwaye use the magic staves over a regular weapon too when I intend on using her as storage capacity, even if if the end they're weaker. Getting them to pick up a shield gets a bit tough too.

Been trying to play around and figure it out , it gets iffy if they decide to use something upon receipt. They're good as using the stronger given armor at least. They have their own preferences?

At a guess: nominal value if it were to be sold. That's what it appears to be. To stop her using a weapon, the only way is to take it off her.

I'm not 100% positive on this, but i believe it has to do with how much value they cost and not the damage done. I believe it also has to do with their attributes. Each companion is better at one thing than another. eg. Their two-handed may be say 75, when their one-handed may be 15. If not on a pc, i've know idea how to find that out on consoles. But, I don't even think that matters as far as what they use, only how much damage they would do with that particular weapon. If a mace is valued at 1000 gold but does 25dmg and you give them say a staff that is valued at 1500 but does 15 elemental dmg, they'll probably switch to the staff. That's what i've noticed anyhow. On the pc there are console commands to force what you want them to wear by taking things out of their inventory and only giving them just one weapon. Some companions may be bugged in that they'll switch over to their default weapons (the ones that are hidden) at times. If that happens, just take back the weapon you gave them and give it back to them. See if any of that helps. WIsh you luck.

-Mush-

The weapon I handed her that she decided not to use is more powerful and more valuble over all and she still didn't change. It was the glass warhammer vs an ebony axe I upgraded. So maybe she just has a two handed preference. But then again, when I got the Elven Greatsword of Scorching (I think that was what it called) and handed it to her a long while back, she didn't use it and it out-did anything she was using at the time in both value and attack power. Now isn't the case, but it was the same 'hmph, have it your way Lydia' moment.

If it is not just a completely bugged mechanic would I say that they are choosing their weapons primarily depended on their situation. My Lydia will choose a bow, a one-handed weapon together with a shield as well as a two-handed weapon if she has these available. I have seen her switching through three different weapons like this against a single opponent while no other opponents were around.

Whatever it is, or how clever the developers thought they would make the NPCs, is the mechanic far from being deterministic. It seems as if the mechanic first picks the weapon type and only then looks at the damage value. Weapon improvements and enchants are looked at at last.

It then also happens that when a bow is drawn will the NPC pick a plain Hunting Bow over an enhanced Ebony Bow. This is obviously broken.

First thing one should do with followers is to remove their base equipment so they cannot fall back to using them. On the PC can this be done with a command on the console and works for weapons and armor, on the 360 and PS can one only try to disarm the follower with a spell or a shout.

Then, once the inventory has been freed of the junk should one pass all available weapons and armor to the follower and see which it chooses. My Lydia will often pick two-handed weapons when they are all Dwarven or Orcish, but when I leave her the choice to pick from all Ebony weapons will she pick the one-handed war axe.

So now I have equipped her with a Dragonplate armor set, a Dragonplate shield, an Ebony bow and an Ebony war axe. The smithing components for the armor literally fall from the sky and it costs nothing to make and it is only second to Daedric armor. She seem to stick to the armor 99% of the time and only rarely puts on a steel helmet that I have given her to carry. When this happens do I take the item back and give her some other junk to carry.

A necklace for an improved healing rate and a ring for more stamina complements it and she has not yet died a single time.

She has truly lived up to her word of protecting me with her life many times.... and I'm usually the one that kills her as she gets in the way of whatever is incoming at the time

Do they decide to equip the little items like that? Rings, necklaces and such? I had her in the steel plate armor of health and various other 'big items', which grant bonuses but never really given the jewlery stuff since I figured they just carry it and not use it.

Yes, I've enchanted rings/amulets and given them to my companions. In every case when i check their inventory after, they had them equipped.

Yes, but not all effects work on them. Best is to read up on the various Wikis on Skyrim. Lydia is good at one-handed combat, heavy armor, blocking and archery, and got some points in two-handed combat, too. She can have up to 660 health but will only go up to 210 stamina, which is why I chose to increase the healing rate and buff the stamina of hers. Once I can give her boots with dual resistances (fire&frost) will she probably be able to kill an ancient dragon on her own. She already kills several Draug Deathlords with her current equipment.

What gives me the most trouble is getting them to correctly dual-wield.

I was trying with that dunmer archer from Whiterun... and I gave her a Daedric Longsword and Daedric Dagger for close encounters... and she kept putting the dagger in the main hand and the sword in the off-hand.
Funnily enough, I also found she favoured the swords over the daggers even when the daggers were upgraded / enchanted to be worth more. As such, I'm not entirely convinced that value is the be-all and end-all...

... And then the bow issue.
But everyone knows that.
I give her a Daedric Bow... she equips it. We get into combat, and she switches out her FAR weaker default bow and uses that instead. >_<;;

I've seen a few followers use what there good at. And some use a Ebony War Axe, when they had a Enchanted Sword.

I had Ilia (darktower lady) She starts with dagger, I gave her a Para dagger, which she never uses. She'll hit stuff with Iron dagger...lol.
I also catch her switching to Tavern Clothes (she wears a hood mages robe) and -20% Destruct spell cost. She wont equip it 99% of the time.

But randomly I'll catch her put circlet on and the Tavern clothes and rock them for a day. Then one loading cell later she has robes back on.

They will even pick their (invisible in inventory) hunting bow over a smithed hunting bow.