This system is based on DotA recipes system which exists for the purpose of allowing a player to get cheap early game items and then combine them together to open up inventory space and make them into more powerful end game item. It's a good system because otherwise players would more often save up a ton of money to get the "best" items right away without spending money on weaker items. This leads to an all-or-nothing item game that really isn't that fun.
The problem with the DotA item system is that it doesn't flow logically since it was built over a long time. Thankfully, Demigod is a brand new game with no real item system in place which should allow for a system with the complexity and depth of the DotA system but with a more logical progression and easier usage.
Let's start by looking at what stats can be put on items in Demigod:
Offensive stats
+ Damage
% Attack Speed increase
% Critical strike chance increase
% Damage Proc (currently Lightning Strike)
Area effect to damage (seems to have been taken out)
Defensive stats
+ Health
+ Health regeneration
+ Armor
% Evasion
Hybrid stats
+ Mana
+ Mana regeneration
% Abilities cooldown time decrease
% Movement speed increase
Thorns Damage
So overall, the modifiable stats in Demigod are a lot simpler than in DotA, where along with all of the above you've got strenght, agility and intellect which would do a combination of the above effects. Some effects from DotA could be added to Demigod as well. One that immediatly comes to mind is % ability damage decrease as a defensive stat - allow players to equip items that would reduce the damage they take from other Demigods abilities.
In Demigod, you have 5 available item slots to fill with items from either of 6 shops: Boots, Gloves, Breastplates, Helms, Rings, Artifacts - unfortunatly the current item categories have no impact on gamplay since you can wear 4 breastplates and 2 helms at once if you feel like it. All items could be replaced with the red blobs of artifacts and it would end up making more sense.
Which is why Demigod should take a page from RPG style equipment screens instead of the DotA item slots and limit the player to 1 item of each type at a time - here is an example

Simple layout, you can equip 1 helm, 1 breastplate, 1 set of gloves, 1 set of boots and 2 artifacts (more on these later).
Now to get back to the earlier idea of cheap early game items transforming into good late game items, there's various ways to go about it, a system such as the one Innociv posted or a more DotA like combination system. This combination would be somewhat simpler than DotA's because it is limited by the smaller number of variable stats in Demigod and by the RPG-like equipment system.
Simply put, you assign certain stats to certain item slots, preferably a mix of defensive or offensive with an item type specific Hybrid one. Here is an example of how you could allocate these stats across the various item types. These are completely arbitrary, but in this example you would have 5 of each item, each with the base hybrid stats, then one with each of the damage and offensive stats, and one with a higher value of the hybrid one. See the screenshot bellow for detail
Helms
Base Hybrid Stat: Mana and Mana regen
Offensive: + damage or % damage procs
Defensive: + health and health regen or % ability damage reduction
Armors
Base Hybrid Stat: Thorns
Offensive: Area effect to damage or % critical strike
Defensive: Health and health regen or +armor
Gloves
Base Hybrid Stat: % ability cooldown reduction
Offensive: % attack speed increase or % damage procs
Defensive: % evasion or % ability damage reduction
Boots
Base Hybrid Stat: % movement speed increase
Offensive: + damage or + % critical chance
Defensive: % evasion or + armor
And each of the shops would have a combination system which allows to merge together any two item sold at that shop into an item with the stats of both, so you buy boots with damage and speed, then later go back to the shop and combine them with another pair of boots with damage and and speed, so you get boots with double the damage and double the speed. This already gives 3 different speed values in game: 2X stats + speed boot (slowest) 2X speed boots (fastest) 1 speed + 1 speed + stats boots (medium speed). Here is a rough example of what the interface could look like, the prices are all arbitrary and can be changed - but the combine should obviously cost more than the price of the boot itself.

Added on to this system would be an item power-up system which is very simple: Remove one of the controlable goldmines in the maps, add an upgrade forge in which there is a simple drag and drop interface where you place an item in the slot and for a price you slightly upgrade all of it's stats. This could be repeatable up to X times at an increasing cost.
This would have the benefit of adding another critical point to control other than the artifact shop and give you more places to spend money which gets quite high in later games. And I think we can all agree that the flag control system is quite nice, especially on a map like the 2v2 where you have to go out of your way to get to them and might run into a 1v1 fight with no creeps around.
Now for the artifacts, they wouldn't get any sort of combine or power increase effect because these items should focus on gameplay changing mechanics instead of simple stats increases (though you could have an artifact or two that does this, preferalby focusing on powerful proc effects like stuns). Things like the artifact that gives invisibility, the one that allows to teleport (though it would be nive if it were a fast "blink" like effect, it would have more combat use) or the one that allows you to AoE buff your units and allies are great examples of this already in game. Other ideas taken from DotA could be crowd controls (stuns, silences, slows?) mirror images, maybe a self resurection, a nuke, etc etc.
These plus the achievement powers which, from what I read, will also change the way certain abilities work would greatly change the way you can play any Demigod - which is really needed because of the low numbers of Demigod. Just being able to focus on being a damage dealer through % increases and passives or a nuker by focusing on damage abilities, or a mix of both still isn't much.
Though I know that in any sort of tournament play it'l come down to one and only one "best" build - it always does.
Thank you for your time.