My thoughts on a BOL machine....
I agree with you Tom that your friend should look for another Kenmore, or at least a Whirlpool product. I agree also with the sentiment that the problem is probably not major with her current machine due to the repair quote being so low, relatively speaking. Did the repairman say what he diagnosed the problem to be?
There was a time recently, and perhaps this is still the case, that Whirlpool was building half the top loading washers sold in this country. The DD design goes back to the early 80s, yours in fact is an 88 model, though the machine may be several years newer than that. When a design lasts so long in production, it's not because its junk as compared to the rest of the marketplace. I much prefer belt-drive WP/Kenmore products, however I realize that due to their age not every average consumer would consider using one, so I recommend top loading WPs every time I get asked.
In addition to Kenmore and Whirlpool, there are clones out there with the Roper, Inglis, Estate, Amana, Maytag, and I think Admiral names on them. At Lowes is a probably unbeatable two-speed Roper, still with a porcelain coated steel basket, super capacity, decent cycles and temps/levels for less than $280 (here in Charlotte anyway). That machine is far from BOL and it should easily last the time frame your friend mentions. Since she's looking for BOL, I presume she's very price sensitive. I love quality built stuff, and the Speed Queen sounds like a deal for the money, but it may be a bit pricey for her. Still cheaper yet is a true BOL Roper for $199 at a local appliance store, and I would much rather own this vs. a comparably priced Frig. or GE etc. (just my own opinion folks!).
I don't know the history of your friend's Kenmore, but I think I'd consider fixing it. Do you think it may need just a coupler? That is a repair a lot of do-it-yourselfers could do themselves and save mega-dollars.