Tl;dr: Bringing Intelligence up to 20 is necessary at some point, but doesn't have to be immediately. War Caster and Resilient are both great options and will help you maintain concentration. Heavy Armour Master and Lucky are also great options but won't have much impact on concentration. None of the other feats seem particularly worthwhile for you. Personally, I'd probably take War Caster and then decide between Heavy Armour Master and bringing my Int up to 20.
I'll address your points one by one, starting with:
Bringing Intelligence up to 20
You definitely want to bring your Int to max at some point, because it controls your primary means of attack. The question is whether you want to get it now, or leave it till later. With that in mind, consider what 2 more points of Int actually provide:
- +1 to hit with your spell attack
- +1 to DCs for your spell saves
- +1 to Int saves
- +1 to Int checks
- +1 to number of spells you can prepare
None of these are huge benefits in the short-term. Long-term, this makes you 5% more effective at most of the things you specialise in. (Warning: gross over-generalisation.)
War Caster
You don't intend to have both hands full, so this only gives you advantage on Con saving throws to maintain concentration, and spells for opportunity attacks. However, both of these are strong benefits. Advantage to maintain concentration needs no explanation, but spell opportunity attacks might not be as clear.
To start with, you're planning to be in the thick of the melee, so you should be getting opportunity attacks often enough to want to use them. Unfortunately, you aren't planning on using a weapon, and your signature spell Vampiric Touch won't work with opportunity attacks. (It specifically requires an action on your turn to make an attack with it.) So in your present state, you have no use for opportunity attacks. Giving yourself an option could pay off.
On the other hand, opportunity attacks aren't the only reaction, and you might want to save your reaction for spells like Shield or Counterspell. That said, these require resources, whereas with War Caster you can always use your opportunity attack to cast a cantrip.
Heavy Armour Master
As you say, the Strength is negligible where you're concerned, so the question is the damage reduction. This is a great option for someone planning to be in melee - 3 damage off every non-magical weapon attack adds up to be a lot of damage very quickly.
For concentration, it's a bit weird. It won't affect the vast majority of Con saving throws you make to maintain concentration, since the damage has to be at least 22 for it to have an impact. (Many high-damage attacks are magical, and it won't affect those either.) When it does affect the DC, it will only reduce it by 1-2. So as an option to help you maintain concentration it's a bit of a bust. It's still a great defensive option though.
Keen Mind
I can't say any of these features have ever come up in a game I've played. They're all highly situational and I can't find anything to redeem this feat other than "it's kind of cool, I guess".
Linguist
3 languages is pretty good, but there are spells that take care of this sort of thing for you. I've never seen a situation in game where a cipher would've been handy. If you're looking for a way to send messages to someone that no one can intercept, there are a variety of spells to achieve that.
Lucky
3 rerolls a day that you can choose to spend once you've seen the result? This is huge. You can spend them on an attacker's roll against you? That is just nuts. If you get critted, you can just wipe it away. Note that its benefits aren't nearly as potent on rolls that you make with advantage, but they're even more helpful on rolls you make with disadvantage. As far as concentration is concerned, you can fix a failed concentration roll 3 times a day, but you're likely to be making a lot of them, so something that isn't a scare resource and benefits all of them is probably worth getting.
Observant
This feat is frighteningly good. Just not for you. For a character who's maxing Wisdom and has proficiency in Perception, this makes them a portable radar station. (One of my players, the cleric, took Observant and has been making my life hell spotting every stealthed enemy and trap a mile away.) For you, it will make you almost as good as they were without it. If your party lacks a "sensor" character, it could be worth considering. Assuming you have one, I don't recommend this.
Everything in the above paragraph is based on Perception, without considering Investigation. That's because Perception is used far more often than Investigation, and passive Perception is used almost constantly whereas passive Investigation rarely sees the light of day. This can vary from campaign to campaign, obviously, but in general Perception is better.
Lip-reading is a cool little feature that I have seen used several times in intrigue situations, but I don't think it's worth spending a feat on.
Resilient
For maintaining concentration, this is pretty strong. More reliable than the advantage provided by War Caster, and scales better as you level. As well as that, though, there are a lot of effects that trigger Con saving throws. Most diseases, poisons, many spells, and a lot of monster effects. There's a saving throw for every stat, but Con, Dex, and Wis are still the most common, so this is definitely worth your consideration.
Ritual Caster
First and foremost, you have every benefit of Ritual Caster (as far as spells on the Wizard list are concerned.) Second, the benefits of this particular feat are highly dependent on what scrolls the DM feels like tossing your way. So the only quantifiable benefit of this spell for you is learning 2 1st-level spells from another class's spell list. Unless you can see something you really want (I can't), I can't recommend this.
Besides that, most rituals are either available to the Wizard or can be effectively duplicated by the Wizard. The only one I can see that you can't produce for yourself (barring a Wish, obviously), is Silence, which I can't quite see how you would use as a ritual. (It's more of a combat spell, and the 10-minute casting time is inconvenient in combat.)
I have few ideas based on your criteria but the best option really boils down to the situations the party finds themselves in and how the party functions in combat. Because of that, I'll list some options and give the pros and cons of each.
Idea 1: Monk 17/Ranger 3
(Take Hunter's Mark, Dueling, and either Colossus Slayer or Horde Breaker)
Possible Damage: 4d10+4d6+[1d8(CS)]+24 for 3 hours (Attack, Extra Attack, Hunters Mark x4 (each weapon attack) [, and Colossus Slayer])
Pros:
- Up to 3 hours of use split into up to 1 hour increments 3 times per long rest.
- Even if you are out of ki and Hunter's Mark uses, you still get Dueling bonus to weapon damage.
- Bonus 1d8 if target is under its max HP, even if you don't have ki or Hunter's Mark. (If you take Colossus Slayer)
- You can hit 1 additional target for 1d10+7 each. (If you take Horde Breaker)
- Ranger class features are nice to have for adventuring.
- d10 hit die
Cons:
- Hunter's mark only recharges on Long Rests and might be ended earlier than intended if you fail a concentration check.
- You don't get the full effect on the first round. You need to use your bonus action to cast Hunter's mark which means you don't get your bonus unarmed or Flurry because it requires a bonus action, only your 2 attacks +2d6 (+1d8 if you have CS and target is below max HP).
- Every time your mark dies, you have to use your next bonus action to pick another mark.
- Because Hunter's Mark is a concentration spell, you risk losing the effect if you take damage.
Idea 2: Monk 17/Fighter 3
(Take Dueling and Battle Master or Champion)
Possible Damage: 4d10+1d8+24 up to 4 times between rests or you run out of ki. (Attack,Extra Attack, Flurry of Blows, 1 damage dealing maneuver)
Pros:
- Superiority Die and Ki recharge on a short rest.
- Maneuvers have utility effects (Battle Master)
- Dueling
- Action Surge gives high burst potential once per short rest. (6d10+1d8+38)
- Second Wind (though 4-13 HP isn't that much at level 20)
- Proficiency with 1 artisan's tools. (Battle Master)
- d10 hit die
- Increased Crit Range (Champion)
Cons:
- You only get 4 rounds of bonus 1d8 damage.
Idea 3: Monk 17/Rogue 3
(Take Assassin or Swashbuckler)
Damage Potential: 4d10+2d6+20 (Attack, Extra Attack, Flurry of Blows, Sneak Attack)
Pros:
- Bonus damage not reliant on recharge. Only have to worry about ki
- If you don't have advantage, you can still sneak attack the target if one of your other party members is in melee range.
- Expertise (double proficiency bonus in 2 skills)
- (Swashbuckler) Sneak attack if no one else is within 5 ft of you.
- (Swashbuckler) Add your Charisma to initiative (if you want)
- (Swashbuckler) Your target can't make AoO against you.
- (Assassin) Crit on surprised target for 8d10+4d6+20
- (Assassin) Proficiency with Poisoner's Kit for potentially more damage or other utilities.
Cons:
- Smaller hit die (d8).
- Smaller minimum damage potential.
- Monk weapon must be finesse. (Not too bad, just limits weapon choice.)
- Can't sneak attack with unarmed strike
- Very reliant on party placement
Best Answer
At the time of the OP
Even before Xanathar's Guide to Everything, you could get 5 saves by having your rogue acquire a Transmuter's Stone with the Constitution save benefit from a School of Transmutation Wizard (PH 119).
A human (variant) wizard could actually get all 6 saves at level 1 with a Shapechange scroll (albeit temporarily). Intelligence from the Wizard class proficiencies, Strength from the Resilient feat, and the other 4 from one of the Dragon Wyrmlings (MM 106).
Now
With the new Xanathar's Guide subclasses, a character can acquire all 6 saves by himself without any form changes. These are the necessary classes in order:
1. Ranger (Gloom Stalker) 7
This provides the Strength and Dexterity saves from Ranger, Charisma saves from Resilient (taken at 4), and Wisdom saves from Iron Mind (XGtE 42).
2. Fighter (Samurai) 7
This multi-class gives Intelligence saves via the Elegant Courtier feature (XGtE 31)
3. Wizard (School of Transmutation) 6
Just like before, the Transmuter's Stone gives Constitution saves.