What I've done in my Hoard of the Dragon Queen Campaign is to have "the local clerics/priests" resurrect dead characters in exchange for the assistance the party is providing the town. So, if your characters are helping the city (I haven't read the starter set, avoiding spoilers), then local authorities could provide healing in exchange. This means, however, that the character has to sit out the next mission (so, they'll have to roll a temp or something--temp could be a generic NPC, like a soldier from the local militia, sent to help the party; or they could roll another character, though it has to be a different race and class than the one that died).
If they actually have the resources to pay for the spell (as per the Starter Set Guidelines), then the character doesn't have to sit out. I think it's a fair 'penalty' without trivializing death.
More specifically, Neverwinter has traditionally been one of the largest (if not the largest) cities in the Sword Coast--undoubtedly such a service is available. The cost would be at least 500 gp, given the material components required to cast Raise Dead.
Then again, the city was presumably destroyed in the 4E lore; without having read the starter set info (I don't want to spoil it), I'm not sure about the current state of the city. This wouldn't be a random NPC; however, it need not be a "high-level cleric--" a L9 Cleric can cast Raise Dead.
About hiring spellcasters, from the PHB, p. 159:
Hiring someone to cast a relatively common spell of 1st or 2nd level, such as cure wounds or identify, is easy enough in a city or town, and might cost 10 to 50 gold pieces (plus the cost of any expensive material components). Finding someone able and willing to cast a higher-level spell might involve traveling to a large city, perhaps one with a university or prominent temple. Once found, the spellcaster might ask for a service instead of payment— the kind of service that only adventurers can provide, such as retrieving a rare item from a dangerous locale or traversing a monster- infested wilderness to deliver something important to a distant settlement.
I'm unsure if this answers your question; as I'm not sure if you want the enemies, descriptions, and all that to be generated as well. Assuming not, I have a non-generated solution that may work..
Node Mapping
For each room you want in the dungeon, put a small or large (or huge) circle down. Connect it with other nodes as desired; call these "links"
When you find a location that you want a particular look/description
for (such as a staircase) simply mark that node.
Build yourself a list of descriptions/rooms/encounters; enough to cover ~75% of the dungeon. (If you think they'll explore everything, fill in the last sections with cave-ins/collapses, submerged passages, the occasional bit of loot; or finally.. just build a list of 100% instead.)
When your players travel a link to a node, describe the first link on your list. When they get to the other side (a room, intersection, etc) then describe the first node on your list.
While this removes some choice from the players (they can choose to avoid certain encounters or whatnot by turning around and taking another path) because their choices will always occur in the same order you a vast amount of control of the order they face encounters, and that they are guaranteed to run into every interesting thing you want them to in the dungeon.
As your players traverse the area they will build your dungeon for you. Simply mark down each list item on the node as they discover it (in case you need to give them the same description as before or remember key details later.)
I suggest this because people often think they need a map that is perfectly defined by size and even placement; but the above actually just has you define rooms and passages rather than a dungeon. Why? Because that's all that you'll end up describing anyway; and it gives you amazing control over the encounters they will face or bypass and in the order that you want them to discover those things.
Your players will think they're wandering a well-made pre-built dungeon with each hazard, discovery, etc well defined and will recognize where they are at by your descriptions; even though you just have some circles and lines on a piece of paper.
What does this have to do with stairs and multiple levels?
Stairs are simply a bottleneck in your node map, and a description of a link. Done and done. Stairs with an encounter are no different (other than mechanically) than a passageway or node with an encounter. Simply mark your "S" node on the map, make it so the second half of your node map have to go through the stairs (or not!) and you'll have a map.
I ran a ~20 node map recently to great effect. The players recognized areas they'd been, remembered where hazards were when they were leaving, knew roughly how things were connected and were none the wiser I just walked them down a list with a mere TWO pre-defined main rooms and everything else being defined as they went.
This made the dungeon have a predictable length and a balanced set of encounters no matter what the players actually chose.
While I will, now and then, design something to be very logical so its description from the outside matches with the inside, that really only matters for marking a couple key rooms down on your node-map.
I highly recommend this approach if you're willing to build your rooms/encounters still. Apologies if you were also looking for the encounters to be generated.
But I want it to line up sq-footage wise!
Fine. For each node define a second node above it of similar size and shape. When your players are upstairs, use the "upstairs" version of the node. Badda bing.
Once again, they build your dungeon for you and you only fill in the blanks; you get to make sure they hit (at minimum) the encounters you were excited about; and if they explore you have a bunch of stuff ready to fill in the blanks.
Upon request, here is a partial example:
Side note: Sadly it seems B1 was lost in translation despite my extreme MsPaint skills. B1 is simply another description of whatever large/interesting area you'd want and some kind of interaction with the players, such as an encounter, puzzle, or hazard.
Note: if you change the scenery in your descriptions you can easily (such as in my case) turn this from a tunnel network to a temple as well. If you want to improve the map, simply add some static (or dynamic!) secret passages, and have some of your nodes (or links) have additional passages that are dead-ends but have something interesting in them (like an optional encounter, a skill challenge or puzzle to get loot, or even just visibility into a future portion of the map.)
Best Answer
The 5000 gp diamond is the only diamond on any of the DMG treasure tables.
As you have observed, the only diamond on the Gemstones tables in the DMG is the diamond appearing on the 5000 gp Gemstones table. If you are strictly using the treasure tables in Chapter 7 for loot, then you will not be giving the players any diamonds unless you hit on the 5000 gp diamond. However, it should be simple enough to swap in diamonds for some of lesser-value gems found in the gem hoards and caches. It is worth echoing here the introduction to the Dungeon Master's Guide:
There is simply no rule which restricts the DM to only using the treasure hoard tables, so there is no sense in which even a strict rules-as-written approach to running the game should prevent you from just giving out some lower tier diamonds to your players. That said, treasure hoards are not the only canonical source of gemstones in the world.
You let them buy what they need at the market.
Gemstones are like large-denomination bills in the economy of D&D 5e:
Fortunately, as the Dungeon Master, you are not restricted to following the written guidance and rules to the letter, in fact, if you did, your world would be woefully incomplete. Gems are "trade goods" in the economy of D&D 5e, and any sufficiently large town or city is certain to have a shopkeeper that deals in precious gems. If a cleric needs a 300 gp diamond for revivify, they should typically be able to find one at the Waterdeep Zales outlet for 300 gp.