There are some basic "recipes" to follow if you want to create elixirs:
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Critter + Monster Part = Elixir
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Critter + Monster Part + Food/Spice = Elixir
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Critter WITHOUT Monster Part or vice versa = Dubious Food
The rules you have to follow with these are:
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Never mix different effects. For regular food, mixing effects still produces an edible meal but cancels out the effects. For elixirs, this always yields Dubious Food.
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Dragon Parts are NOT Monster Parts! The game rather treats them like seasoning/spice so you can use them in all dishes and elixirs, like you would use salt. If you want to use a Dragon Part in an elixir, you need to also add an actual Monster Part.
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Adding more critters strengthens the effect and also extends the duration. Adding more Monster Parts extends the duration but doesn't do much for the effect.
Now for the in-depth version:
Just like there are strong and weak ingredients for regular cooking, some of the critters are simply better than others. If you, for example, cook a monster part with 4 Summerwing Butterflies, you will get a lv.1 Cold Res elixir, whereas 3 Warm Darners are already enough to boost the effect to lv.2
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Winterwing, Summerwing and Thunderwing Butterfly are "weak". Cold Darner, Warm Darner and Electric Darner are their stronger counterparts.
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Fireproof Lizards are the weak Flame Res critters. Smotherwing Butterflies are stronger, unlike the other butterflies in the game which are "weak".
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For speed boosts, Hightail Lizards are weak and Hot-Footed Frogs are strong
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For stamina restoration, Restless Crickets are weak and Energetic Rhino Beetles noticably stronger.
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The Sunset Firefly, Tireless Frog, Hearty Lizard, Bladed Rhino Beetle and Rugged Rhino Beetle do not have critter-counterparts.
Using stronger critters will produce stronger effects, but as a tradeoff, the stronger critters are usually more rare than their weaker counterparts. This is especially noticable with the Restless Crickets (weak and basically everywhere) and Energetic Rhino Beetle (good luck finding one, but 2 of them already refill all three rings of stamina)
Dragon Parts are still NOT Monster Parts, but you can add them to an elixir if you ALSO use an actual Monster Part. For the best result, use Dragon Horn Shards as they guarantee a 30 minute duration and random extra crit on top, like +3 red hearts or +1 level of effect.
Monster Parts:
As for how many and which monster parts you should use, this depends mainly on what you want to do with the elixir in question - consume it or sell it.
Monster parts have 3 "tiers" - common, uncommon and rare.
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"Common" ingredients (Tier 1) create elixirs with a 3:40 minute base duration, and adding more of those will boost the duration by 1:10 min.
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"Uncommon" parts (Tier 2) offer a base duration of 4:20 instead and adding more yields an additional 1:50 min. per part.
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"Rare" parts (Tier 3) offer a base duration of 5:40 minutes and offer an additional boost of 3:10 min per added part.
Each monster has at least two different tiers, and those tiers are the ONLY important part for the duration. It doesn't matter in the slightest which MONSTER those parts came from: as long as they're in the same tier, they will offer the same duration. A boring Keese Eyeball will give you the same effect as a Giant Ancient Core, as both are Tier 3 ingredients, so it doesn't make much sense to waste the valuable stuff on elixirs that you plan to consume.
...and if the elixir doesn't have a timed buff in the first place (like stamina restoring elixirs) then the monster type and part do not matter at all. For those, always use the cheapest, most common monster parts you have, like Chuchu Jelly or regular Keese Wings.
The tiers:
Monster Type | Tier 1 | Tier 2 | Tier 3 |
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Chuchu | Regular Jelly | Elemental Jelly | (none) |
Keese | Regular Wings | Elemental Wings | Eyeball |
Octorok | Tentacle & Balloon | Eyeball | (none) |
Bokoblin & Moblin | Horn | Fang | Guts |
Lizalfoes | Horn | Talon | Regular & Elemental Tails |
Lynel | Horn | Hoof | Guts |
Molduga | (none) | Fin | Guts |
Hinox | Toenail | Tooth | Guts |
Guardian | Screws & Springs | Gears & Shafts | Regular & Giant Ancient Core |
The only time you need to pay attention to which MONSTER the part came from, is when you're planning to sell the elixir - because the selling price is determined by the value of the raw ingredients, completely independendly from the effect the elixir has. A meal/elixir made with 5 items at once sells for approximately 2.8 times the value of all raw ingredients combined, and if you use more "valuable" monster parts, then the elixir will sell for more - even if the effect is total garbage.
Example:
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An elixir made with 1x Restless Cricket and 4x Bokoblin Horns will heal the bare minimum of stamina and sell for 40 rupees.
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An elixir made with 1x Restless Cricket and 4x Lynel Guts still only heals the bare minimum of stamina so it does the exact same thing as the other elixir - but is sells for 2,250 rupees, simply because the Lynel Guts are worth more cash.
If you cook elixirs to SELL them, always use the most valuable monster parts and always cook 1 random critter and 4 monster parts together as this will give you the highest multiplier. This can be a convenient and easy way to make some extra cash, as the monster parts you already own basically get thrice as valuable just because you tossed them into a pot.
Oh and keep in mind that you can turn regular Chuchu Jelly into elemental jelly by setting it on fire, freezing it or zapping it, which boosts their effect by one tier AND makes them slightly more valuable.
If you want to experiment around a bit with elixir ingredients without wasting your own resources: Here is a neat cooking simulator for the game. ;)
EDIT: I'm stupid and didn't read which community this was. Disregard!