Hi Krusty, there has been at least one other discussion on which Honda to use, but it was a brief reference and there's no harm in bringing it up so that more people get their chance to express their view and experience.
Reel mowers do not consume much power, except to drag themselves up hill: they are usually pretty heavy compared with a rotary. You didn't tell us the width of your father-in-law's SB45, but if it is 14" or 17", I suggest a Honda GX120. It is rated at 3.5 horsepower net (not gross, as the original engine was). Even for a 20" it has more power than the original engine that was fitted. It also runs vastly better than a Kirby Lauson, I agree with your father-in-law on that subject. Its disadvantage is its price: you can buy a side valve Briggs for much less, and while a Briggs is no Honda, it does run far better than a Kirby Lauson.
If you are concerned to get the best value engine possible, I suggest you get a chonda (an unlicensed Chinese copy of a Honda). They are even cheaper than a Briggs, and run as well as a Honda, while the base engine has Honda's superior durability as well, but the disadvantage comes in the bolt-on parts of the engine. The starter, governor plate (external speed control mechanism), air cleaner and muffler usually do not exactly follow the Honda design, and may give trouble. Furthermore parts may not be easy to get. The base engine parts are not a problem, since they are interchangeable with Honda parts, but this is not true of some of the other parts. (Of course you can nearly aways fit the whole Honda system, but unless you had a junked Honda on hand, that would be unaffordable.)
You will find that the cheapest, easiest-to-find replacement engine is a chonda version of the Honda GX160. This is a bigger engine than the GX120, rated at about 4.5 hp (gross is 5.5 hp, and net is not given in the specification). The chonda version has the same advantages and disadvantages as the chonda version of the GX120. More importantly, it is a bulkier and heavier engine than the GX120 (though not as heavy or bulky as the Kirby Lauson) and you might find it fouls the handlebar, so you should check this before making a purchase.
Remember, while all the engines mentioned will bolt straight onto your engine mountings and the crankshaft extension (PTO) comes out in the right place, the length, diameter, and even shape of the PTO is subject to multiple options, so you have to choose the right one. Two sizes of engine-side clutch-half are available as SB spare parts (5/8" and 3/4") but this won't help if you buy a metric shaft-size, or one that is tapered, or doesn't have a keyway, or even has an external thread. Understand what you are buying before you buy it.
