While having a belayer that is much lighter than the climber is clearly not ideal, especially on lead, it isn't insurmountable.
A famous climbing couple from days of yore was comprised of Lynn Hill and John Long. Lynn Hill is 110 pounds and while I can't find any exact numbers John is known as a big muscular guy. While they were clearly elite climbers it goes to show you that radically different sizes can be good partners. On a more anecdotal level my first climbing partner was a body builder and I was a fairly scrawny guy. I never asked his weight but I'm sure he outweighed my ~170 lbs by at least 30 to 40 lbs and we never had any problems. Also a more contemporary example would be Ammon McNeely and his girlfriend's second ascent of Wings of Steel. I don't know their weight difference but Ammon is clearly bigger than her and he regularly took massive falls on that route day after vicious day. Ammon is a terrific guy and a world class climber. Kait on the other hand certainly has experience but she isn't nearly at Ammon's level.
According to a write up on the Edelrid Ohm product that Michael also linked to in his answer:
The newest educational standard from the German Alpine Club (Deutsche
Alpenverein aka DAV) suggests there should be no more than a 10kg (22
pound) difference between the belayer and climber. This number has
been reducing over the years as the DAV continues to study the belay
habits of climbers to ensure a safe experience.
For what it’s worth, I weigh less than 120 pounds. In the last 7
years, my main climbing partner, Andreas, has weighed between 160 and
175. This is a +35-48% difference – which is higher than even the old European standards recommend. I can actually tell if he’s gained or
lost weight in how it feels to belay him. When he’s on the heavier
side of that spectrum, even during a smooth lower, I can be lifted off
the ground. Whenever he falls, my feet come off the ground.
These quotes would suggest that my climbing partner was oversized for me per DAV guidelines. We made it work, and people with a really big disparity (48% difference) can make it work too. So I don't think people should dogmatically dismiss a good climbing partner just because a weight disparity exists.
Now that we have that out of the way what can we do when a weight disparity exists?
Increase friction in the system
If you read the review I linked above you can see that the Ohm is basically a device that increases rope drag when the system is loaded. The basic principle at work is the more friction/drag in the system the less upward force is applied on the belayer so the less likely they are to be lifted off the ground. You can also see this technique in climbing gyms that use a round bar as an anchor point that the rope is wrapped around for their top ropes. The more wraps, the more friction in the system and the less force is transmitted to the belayer.
Anecdotally I lead a multi-pitch climb where at the top of one pitch there was so much rope drag (because I didn't use long enough slings) in the system that I had to pull with all my might on the rope in order to get any slack. Luckily by that point in the pitch it was really low angle slab, so it was a pain but not unrecoverable. In any case due to all the rope drag I bet my 4 year old daughter would have been able to hold me if I would have fallen.
Anchor the belayer/belay from an anchor
If we don't want something to go somewhere you anchor it to an immovable object. Depending on the gear you have and the surroundings you can probably find a way to tether the belayer to the ground. Once they're anchored to the ground, or belaying from an anchor, who cares what the weight disparity is as long as they can work the belay device because then their effective weight is orders of magnitude larger than the climber's weight with a proper anchor. Keep in mind too that even if they're too weak to work a tube device there are auto-locking devices like the Grigri.
It is important to keep in mind that these are the default options everyone has when doing multi-pitch climbing, regardless of weight differentials, because the belayer must be anchored while belaying. When doing multi-pitch climbing it is considered a best practice for the leader to belay the second directly from the anchor. Here is one out of a bunch of resources on the internet covering this technique that specifically mentions disparate weights. Directly belaying a leader from an anchor doesn't have as widespread support. Two common concerns are increased forces on the system due to an unyielding anchor, and that anchors aren't typically setup to counteract upward forces. The first complaint in my opinion is mostly valid when your protection is questionable. The second complaint is a bit of a red herring because anchors should be built to handle any forces they may be subjected to. Typically this is a downward pull, but if you are planning to belay off an anchor then you should obviously build an anchor that can support that. Personally I've almost always just belayed off my harness, but I've occasionally belayed a second directly off the anchor, and never belayed a leader directly off an anchor. That being said I've never felt like the situation warranted me belaying a leader directly off an anchor. However, if I felt like the situation warranted this then I'd strongly consider it.
Accept the fact that the belayer may be lifted up and the leader may fall farther with a more gentle catch
Being lifted off the ground as a belayer typically isn't the end of the world if the belayer is on the ball. Some belayers even consider this as part of the fun. Of course there are times when this is dangerous for the belayer and/or the leader. This is a factor you'll have to judge for yourself. Also keep in mind that the person taking the lead fall will experience a more gentle fall because some of the force of the fall is being absorbed by the belayer being lifted off the ground. This will also reduce the forces on your climbing protection.
In my personal experience belaying people who are much heavier than myself I've found what generally happens is I get pulled up a couple feet up in the air, then I lower myself down to the ground, and once my feet are back on terra firma I'm able to lower the other climber. Of course your mileage may vary...
Notice I didn't recommend you make your partner wear weights while climbing. Climbing should be fun for everyone, and having to stand around carrying a bunch of weight, or slug a bunch of lead weights up multiple pitches isn't going to be fun for long.