How can I magnify ≤1mm objects onto a display for under $200?

Just one suggestion:

This person is doing fairly precise work while doing their best to squint at details through a tiny handheld magnification lens that is most definitely not up to the task. An external screen is a strong preference, as it will eliminate eyestrain from constant super-focusing.

Using a magnification lens correctly does not require "constant super-focusing" since the focusing distance can be reasonably adjusted. However, depending on the work "handheld" seems like asking for unnecessary constraints. I would strongly recommend looking at common watchmakers' tools here: those usually are implemented as eyewear and are hands-off.

An important advantage of optical tools not involving a screen is that the image has depth, even to some degree (through accommodation) when only using one eye. This is rather helpful for delicate manipulation since you need a good idea where your tool is going to touch the workpiece.

There are a lot of USB microscopes that could do it on the cheap... Image quality probably doesn't need to be photographic in quality. Aliexpress has a number of them- with screens. Examples are here and here.

There are a few considerations here:

How are the objects illuminated?

Small objects typically need quite good illumination to see them properly. Without good light there is little point in magnifying them.

What sort of objects?

Are they solid or in a liquid medium? transparent/translucent/opaque? Do they need to be manipulated (e.g. rotated)? Are the objects mounted on something (e.g. maybe small flaws on a pipe) or can be moved/hand-held?

If objects are solid and opaque I would look at systems such as stereo microscopes (AKA dissecting microscopes), which are designed to look at small objects from above with reflected light. These have long focal distance lenses, which aid in inspection and rotation of objects. These have the added advantage of giving stereo/depth vision (though not usually with a camera). I would use this with an external adjustable position light source, but such microscopes often have built-in light sources too.

Transparent objects might stand up to using a "regular" (compound) microscope such as you pictured. This sort of microscope has a very short focal length and is not designed for manipulation of the objects at all - usually subjects are mounted in liquid on a slide or in a solid medium.

With both of these options you get what you pay for - the lenses are key. Cheap microscopes usually have terrible optics, which wouldn't allow a clear picture of the object, no matter how good the camera attached to it is. The microscope and camera you posted above for $150 will give awful images.

Having said that universities and similar organizations often sell old (but quality) instruments for fairly cheap - look around for second-hand lab supplies on Ebay or similar. I use microscopes regularly for work and good (new) lenses often sell for several hundred dollars apiece, with a fully set up (compound) microscope costing up to about $50,000...quite a bit out of your price-range! Microscope cameras can be pricey too, depending on what you want.

Do they need a screen/camera set-up?

An additional option is a visor with lenses attached (e.g. these ones. I have no affiliation with this company and make no assertion on the quality of product). These are often used by hobbyists and model makers for magnifying small objects such as figurines when painting, or jewelers when fixing rings etc, so are good for small manipulations and give nice stereo vision. I imagine for ~$150 you could get a quite good set of these. This would eliminate the hand-held aspect and the need for screen with consequent lack of depth perception.

How would you mount a camera system?

Having a camera system is all good - but for a good quality image you need good quality mounts and lighting. The mounts keep the camera and lenses stable, while the lighting gives the option of decent exposure times/apertures/iso etc.

lenses?

Note that the camera system "zoom" lenses are typically for magnifying objects that are far away (like a telescope) - what you seem to want is a macro lens for looking at small objects, close up. For this quality of lens is again the key (and camera sensor too...) to the image quality. Poor lenses can give you chromatic aberration, and/or distortion, which may or may not be critical for your friend, but are certainly things to consider.

I have a USB microscope similar to this one. (I won't post a link otherwise this might seem like an advert - there are many similar products and I think they're probably all just as good.)

You can find similar products sold as microscopes, otoscopes or endoscopes, and adding "USB" to the search terms will tend to help.

The image quality is surprisingly good for something in the US$20-$30 range. It has a manual focus ring (no autofocus, but for macro work you probably don't want that) and a built-in light. It can focus anywhere from touching the lens up to I guess 20cm or so away, so there's no need to prepare slides or anything. The computer sees it as a normal USB web cam, so I never had any software issues with it at all. I use the built-in app "Photo Booth" on a Mac and it just appears in the drop-down list of cameras.

If this is for photography (i.e. if you want to produce "good" images) then you might want to spend more, but for your stated goal of clearly resolving millimetre-sized objects it might be more than good enough.

Here's an image to give you an idea. Not a professional quality image, but clearly legible. These are Japanese rice fish eggs. I don't know their exact size but they were quite a bit smaller than 1mm. I believe the small dots inside are the blastocysts (the first few stages of cell division), so I guess they're pretty small.

Frame challenge:

As someone who sometimes does assembly work on a similar scale, I find significant disadvantages to your proposed approach, to the extent that I'm not sure it would be a good surprise.

A stereo microscope is probably the best tool for the job - unlike a realistic camera-based solution it maintains binocular vision and depth perception. The one I use in work can divert the light from one eyepiece to a DSLR; I tried hooking it up to a monitor once, and it was unhelpful (though this was partly an issue of dynamic range with a mixture of dark parts and very shiny ones catching the light). These exist, new, within your budget.

I sometimes use a jeweller's loupe (eyepiece magnifier) but don't find them comfortable for long periods. That's higher magnification than most handheld magnifiers and hands-free.

Alternatively there are illuminated, head-mounted magnifiers, some of which have a lens for each eye. They can be quite cheap or much more expensive

I'd assume he would want his/her hands free from holding the scope while doing precise work. In that case, he/she'll need a scope with a form-factor similar below. This is an Andonstar 4.3" 1080P Display Digital Microscope Magnifier. Costs around $69.

Here's a video of it at work:

https://ne-np.facebook.com/martview/videos/andonstar-ad106s-43inch-display-microscope-1080p-220x-usb-digital-microscope-cam/2108005559219353/

Likely the best choice is surgical / dental binoculars. I suggest 3.5X however your get headsets that yield higher magnifications. Likely you have seen similar worn by your doctor/dentist.

I purchased this microscope (or one VERY similar to it) a few years ago, and I was very impressed with it. I felt like it was $40 well spent.

https://www.amazon.com/Plugable-Microscope-Flexible-Observation-Magnification/dp/B00XNYXQHE

Frame challenge: A good cellphone might suffice. Yes, you won't get a cellphone with a great camera for less than $200, but there is a high chance that you, your friend, or someone else you know already owns one and that you can borrow it.

This is a picture I took with my Samsung Galaxy S9 in June. A kid had an itching, tiny brown dot on their skin, and it was impossible to identify the cause with the naked eye (and I have good eyes). Zooming in with my smart phone revealed the culprit.

Image hidden in a spoiler tag, since ticks are technically spiders and I don't want to shock anyone. For reference: According to Wikipedia, a diameter of a human hair ranges from 17 μm to 181 μm.

You can get fairly high resolutions by putting your [immobile] bugs on a flat bed scanner and displaying the resultant image on a PC.

With luck, you might get 2400 dpi.

Excellent illumination.