What's The Best Way To Sell These Coins?

I think a lot will depend on what grades these coins are in and how much work the grandmother (or you) is willing to put into it. Are they low grade circulated sets put together from pocket change, or are they high grade AU/UNC sets?

The Lincolns will almost certainly do better if sold piece by piece rather than as a set. If it is a low grade circulated set, I would at least sell the early teens D and S mints, the 14-D, 22-D, 24-D, 26-S, and the 31-S by themselves. She could then sell the rest of the common stuff together. If this is a high grade AU/UNC set, then some closer attention to each individual piece would definitely be warranted.

If you are talking about a bunch of high-end MS stuff, it may be worth sending a bunch of them off to be certified. (This goes for all the coins, not just the Lincolns)

Seeing that they need the money, I doubt she would get them certified. If they are indeed high-grade, auction them off as one entire lot.

You have 4 groups: pull out the proofs of each group and sell them (3 lots), and then the 4 groups (minus the proofs). So you have 7 lots. Sell them at the auction right next to each other so that the same buyer pool are tempted to bid on everything just to have a complete set. That's my amateur opinion.

Good luck!

Selling them off is easier said than done for those with no coin selling experiences. There are way to many problems here that require some carefull examing of what is there. As already noted the grade, condition, of the coins is really important. Some coins regardless of grade have little difference in value but others have massive differences. For example a 1931S Lincoln Cent Changes very little from the lowest grade to the highest. Whereas the 1914D can vary from a few hundred dollars to many thousands by grade.
And it's easy for someone to say sell this or that. However, just where do you sell them? On ebay? Are you familiar with ebay? Other on line web sites? Not a coin store for sure. Do you know how to find a coin show? Could you go there with a list of what you have to sell?
And as already noted if she needs the money, not much chance of sending them in anywhere for certifications either.
Are these coins in Folders, Albums, loose? Also, you really need a list of coins to realize what you have there. For example with Mercury dimes there is also a 42D/41, 21, 21D, Several Micro S Mint marks besides the more famous 45 Micro S.
I mention coin shows since if you did find one you may find a dealer that is willing to help you sell them all off for a slight fee. Or possibly purchase the entire collection. But regardless, you should really figure out what exactly you have and what you think it's all worth.
Since your not a coin dealer you must realize that whoever purchases those coins and if for resale, they too need to make a profit so you would never be offered what they must sell them for. Almost all price listings for coins are based on what you would pay for them, not for you to sell.

Thank you for all the input. I will pass along your advice to the family and see what they want to do. You may see some of these for sale here.

And to asnwer a couple of your questions:

cwb - Other than the proofs, they are low grade sets from circulation with the occasional nicer piece thrown in.

Bad Thad - my mistake. The Lincoln proofs start with 1975.

just carl - no chance for certification. Also, no 42/41 Merc, and only one Micro S - the 1945.