While I prefer other systems for fantasy, I keep up with D&D just in case I have players that want me to run it. Any way, my thoughts
Multiple dice: Personally, I don't feel multiple dice are necessary. OD&D with Chainmail used only d6s before multiple dice were used. I might actually prefer Chainmail's 2d6 rather than d20 roll to hit.
Ability Scores: Keep it. Personally, I would prefer moving to using specific traits similar to Tinyd6 Traits or Asset as in Cortex Plus Smallville. What I am referring to is a strong character taking a trait of Strong modifying any size related traits due to race (e.g. Large, Small, Tiny), an agile character taking Acrobatic and/or evasion traits, a manually dextrous taking Sleight of Hand, Lock Picker, and/or any other trait indicating fine hand/eye coordination, an Intelligent character could have certain traits. Then have a trait grant advantage or modify difficulty on appropriate tasks. Alternately, taking skills could serve the same function. However, since this will not happen, I guess keeping ability scores for tradition sake or fine.
Keeping the current Ability Score method, i would make the folloiwng changes:
Strength and Dex to hit bonuses: I can't recall if this was offical for 0E or Gary's house rule for 0E (circa 2005), but i would implement that Strength does not modify the to-hit roll and Dex does not modify the to-hit roll with missile weapons.
Distinct races/lineages: I am for keeping it, but I would label subraces and half races as lineages or ancestries. Also, I would remove cultural elements out to an additional choice of culture(e.g. Nomad (Hunter/Gatherer), Nomad (Pastoral) Horticulture, Agrarian (choice of Rural, Urban, Special)).
Certain cultures might choose from long or narrow list of specific environmental subtypes (e.g. Arctic, Aquatic, Coastal, Desert, Forest, Grasslands, Hill, Mountain, Jungle, Marsh/Swamp). The culture would grant certain proficiencies or abilities and, may even, determine which background choices are available to the character.
Distinct classes: Another keeper. However, I want even more base classes, class variants, and archetypes/subclasses at level 1. I also want more customization choices at certain level for certain classes- especially for the monk.
Alignment: only for supernatural creatures and divine/infernal powered PCs.
Backgrounds: This is one of my favorite additions from 5e. I think it should stay.
Multi-classing: Yes, if it remains an optional rule. As a DM, I hate making players jump through multi-classing hoops to create common fantasy archetypes which is why I prefer more classes, class variants, and archetypes/subclassess at first level.
Feats: Yes, if they remain optional.
Proficiencies: Since D&D will not move to a trait system similar to Tinyd6 or Smallville's Assets, I am for keeping 5e proficiencies, but there should also include an optional 3e type skill point variant
Levels: Yes, but slow down the leveling
Experience Points: Yes, as long as milestone leveling is an option and there are other options for exp besides killing things
Hit Points: I don't think it would be D&D without hit points. I am also open to alternate methods for determing hit points and for reducing them at higher level. In OD&D, everyone orignally used d6. Later editions, prior to 3e didn't, use hit dice to determine hit after certain levels.
Hit DIce: I am fine with hit dice for healing if this is the reference
Armor Class: Yes, but include an optional Armor as DR variant)
Lists of specific equipment: Yes. However, I would like to see a list of equipment broken down into appropriate eras. Also, move "fantasy" armor (e.g. studded leather) to an optional rule/sidebar and replace with historical armor.
Saving Throws: Yes, keep the three saves. Also, I like the base DC starting at 8 in 5e rather than 10 before modifiers.
Damage Types: Keep. It helps differentiate weapons, spells, and certain monsters.
Conditions: Definitely Keep. I like having different effects as they make combat interesting and lend to creating "story"
List of Spells: I am fine with spell lists. However, I would drop from a future core rule book most of the spells that first appeared beginning in 1e Unearthed Arcana or later editions/supplements. Also, I would create a smaller base list of "shared" cleric spells and add more domain spells to round out a cleric's list.
Deities: Keep deities and pantheons both unique to their specific worlds (although Greyhawk should use Gary's Oerth Deities not Lakofka's Suel Pantheon as default) and as examples that both new or time limited DMs can grab if they choose
Great Wheel Cosmology: Use as an example and as a default for Greyhawk and Forgotten Realms. However, keep it away from Dark Sun and specific other pre-published settings.
World Axis cosmology: Use as an example of cosmology and as a default for Nentir Vale. Keep away from other pre-published settings
Creature types: Definitely keep, but I would replace Fey with Spirit and make Fey a Spirit subtype. Why? Because many cultures have their own versions of Spirits and not all spirits are Fey, exist in the Feywild, or follow the Fey hierarchy. Many of these spirits also have their own abilities and weaknesses.
Challenge ratings: Keep. I personally ignore them, but many DMs find them useful
List of specific magic items: 5e's list or a list in general? Ready made items are always useful
Advantage/disadvantage: In my opinion, this is another of 5e's best additions to D&D. I am for keeping it.
Other: I have already listed a few things from earlier editions that I would like to see (even if as optional rules). I would also keep both Bounded Accuracy and Inspiration.