How do I give gifts to children?

So I found the problem and I am answering for those that may encounter it too. It's certainly some kind of bug.

Problem details: "When trying to give items to the kid (givable items of course) the items in the give menu were grayed out as if the kid was overencumbered and could not hold any more load"

The bold part is actually the key to the solution. In my case the boy was indeed overencumbered. And actualy he carried 478 baskets (??)!! More than 500 weight. Consequently any attempt to give anything to the boy was useless.

The bug, here, is not in the "give items" dialog. It's rather in some other part of the hearthfire plugin that made the boy to have 478 baskets in his possesion. You also cannot inspect the kid's inventory to be able to see what he posesses.

Anyway, the solution here involved, ( as most of the times ) running console commands. Specifically these are the steps

1. Open the console by pressing the ~ key
2. Click on the boy (important step, since you want all the following commands to be executed for the boy). Make sure the boy's refId shows up in the top center of the console.
3. Execute "removeallitems player" (without the quotes). This will remove all items from the boy's inventory and will add them to your inventory.
4. Execute "disable" (this will make the boy dissapear)
5. Execute "enable" (this will make the boy visible again)

I am not sure if the last two steps are actually required.

This hack can also solve similar inventory problems with any other NPC in the game, either in a DLC, in vanilla skyrim or any other mod/patch. (Just open the console, click on the NPC and follow the above commands). Be sure that you give back to the NPC all the items they should have (leaving out all the items they shouldn't have)

For example, I faced a similar problem with Mjoll the lioness. She was my companion. I married her too and later in the game I dismissed her from being a companion, and Serana (from Dawnguard DLC) replaced her as my companion. When I installed UFO mod that allows for more than one companions, I went home to re-reqruit my wife, Mjoll, as my companion again. But this time it seemed that I couldn't trade with her more than a few items, before every trade option was grayed out (meaning she was overencumbered). And I remembered she could always "carry a good bit of gear". Also, I couldn't see anything sucpicious in her inventory whenever I traded with her. She carried the armor I had already given her, and the weapon I had given her too. Exactly the same pieces she had when I first dismissed her.

This is a strange behavior of the skyrim game.

1. A follower's default armor will not show up in their inventory when you trade with them. Yet, it is there and the weight counts.
2. In order to ensure that a companion will always have at least an armor and a weapon, the game will always add the default armor (and maybe a longbow) to the follower every time you hire/reqruit him.

This behavior has the following side effect. When you dismiss and rereqruit a follower, armor pieces will be added each time to their invenotry, encumbering them, but you will never notice, because you won't see anything in their invenotry. Consequently, re-reqreuing the same follower again and again will leave less and less free space for carrying items.

By using the console hack above, clicking on the NPC (while in console mode) and entering "removeallitems player" all of that NPC's items will move to your invenotry, including the hidden pieces you couldn't see before.

When I did that with Mjoll I realized she had five full sets of blades armor in her invenotry(!!) and that was why I couldn't give her any more items.

After stripping her naked, I disabled her, and enabled her again, and she reappeared with her default Blades Armor (one piece this time), and through the trading dialog, I gave her back all her stuff, (except from the five pieces of blades armor of course), including the armor I had made specifically for her. From this point on, I could trade with her again, as I would do initially.

IMHO this is a silly bug. As a programmer I can't imagine a reason for not checking if a follower has already an armor when you hire/reqruit him (and in this case, not adding his default armor again).