Here are some original old school D&D modules that should be easy to find and not cost you an arm or a leg if you buy them online:
I1 The Forbidden City (TSR, 1980): Old school AD&D sandbox setting in a "lost" jungle city inhabited by snakemen (yuan ti), frogmen (bullywugs), and lots of other weird and dangerous creatures. Lots of room for development here and no two adventures will play alike (due to multiple adventure paths and foes).
B4 The Lost City (TSR): This old school original D&D module consists of a pyramid buried in the desert sands which holds various degenerate groups battling each other for control of the underground empire.
X1 Isle of Dread (TSR): Lost island sandbox setting with lots of different adventure ideas
WG4 Lost Temple of Tharizdun (TSR): written by the master himself, EGG, this AD&D adventure takes place in a long abandoned temple to a dark god hidden in a mountain valley. Initial conflict is against monsters who now inhabit the structure, but if the party delves deep enough they will run into Things Best Left Undisturbed.....!
Axe of the Dwarvish Lords (TSR): This AD&D supermodule written in the 90s takes place in an long abandoned Dwarvish citadel now inhabited by an army of goblins.
Gates of Firestorm Peak (TSR): This 90s AD&D module takes place in a weird, otherworldly mountain that has connections to a "Far Realm" of madness and insanity. Sounds like it would fit right in with Raggi's stuff!
Not as easy to acquire are the classic Judges Guild modules Caverns of Thracia and Dark Tower (however, the 3.5 reprints should be a lot easier to find and are backwards compatible). Both are great and concern ancient, underground empires with lots of evil things lurking about.
1481 DR
If you look at page 30–31, under the "Ruins of Thundertree" heading, it mentions the eruption of Mount Hotenow as being thirty years ago. Mount Hotenow erupted in 1451 DR, so that means it would currently be 1481.
What this means for your player is that there is not a king. The current ruler would be Dagult Neverember, who has a son named Renaer. You could skirt your lore around a bit and erase Renaer from existence (he does not feature in LMOP) and simply replace him with your player's character.
This will, however, make things awkward for you if you plan to use those characters in Waterdeep: Dragon Heist however, as Renaer does feature there.
Best Answer
You're in luck, because the power levels of the old Basic D&D games are similar to those of 5e.* I'm going to illustrate my answer with examples from B5 Horror on the Hill, but you can apply the principles to any of your Basic D&D Modules.
1. Replace what monsters and NPCs you can from the 5e Monster Manual
This will save you a lot of work, and will also give you a feel for the differences between the two editions. The 5e Monster Manual is quite comprehensive, although you may have to be a bit creative - so there are no Neanderthals in the MM, but there are "Tribal Warriors".
2. Convert the rest of the monsters and NPCs
As I said, the power levels are fairly similar, so there's not too much work to be done here. The main tasks are:
3. Make sure you understand how Ability Scores and Saving Throws work in 5e, including setting DC, and passive Perception
You can read up on these in the 5e core books you have. Most of what you need to start is in Chapter 7 of the PHB, and the DMG gives more detailed advice if you need it.
So for example this trap from B5:
I would interpret this in terms of a DC 15 Wisdom (Perception) check to spot - so unlikely to be spotted with passive Perception, and then a DC 10 Dexterity saving throw so as not to fall in - about even chances, with high Dexterity making a difference. Note that in 5e both of these rolls mean trying to roll the target number or above on a d20.
4. You can get more detailed with the DMG
An alternative way of dealing with traps in particular would be to take the most similar ones from the examples in the Dungeon Master's Guide, so 'Hidden Pit' (DMG 122) seems the best fit here.
The same goes for magical items - either leave them as they are, or take the nearest alternative in the DMG.
5. Enjoy!
There are other curiosities about B5, for example the (to modern eyes excessive) use of percentages to determine if things will happen, but these can actually make things quite fun for the DM in terms of the unexpected happening. But apart from the above changes, most of which can be made on the fly, you are good to go!
*You can test this by comparing the "Wandering Monsters on the Surface of the Hill" table (B5, 5) to the "Wilderness Encounters" table of the 5e Starter Set (The Lost Mine of Phandelver, 27) - the entries are comparable and in some cases almost identical.