Mom's 60th birthday

How long a visit are you planning?

How many days/nights do you have for this trip?
How does your group intend to get around? Public transit? Hired drivers? Rental cars (this one requires some homework)?
"Tuscany" is a region: what does your mom want to see/do there that would help us drill down to places?
"Amalfi" is a small city. Amalfi Coast is a region so same question: what does she want to see/do there?

We will have 10 full days in Italy (this does not include our flights to and from - so 12 total). We want to stay somewhere in Tuscany that will allow easy day trips to see Florence as this will be the first time for the majority of the group. I'd like to take my mom to see San Gimignano as well. In the Tuscany region she wants to do wine tasting and a cooking class.

Most of the group has visited Positano (just day trips so wanting to go back and fully experience/see more). Of course, we want to get some beach time in and possibly a day trip to Capri. Any recommendations are appreciated for this area as well.

I know it's mom's bday and she should get her wish, but with that many people and that time frame, I would stick to Tuscany. You definitely do not have time for three places. Is everyone wanting to stay together? Maybe a villa somewhere on the train line? Or in Siena, where you can catch the bus to Florence.
Could she possibly go to the Amalfi coast by herself after the group celebration? That would be my bday wish!

Rome would not be a bad route even if you include both places, as transport is convenient.

4 nights in Florence. That is basically 3 full days with possibility of one day trip to San Gimignano & maybe Siena on the same day tour (you should probably use a private van tour for ease of travel with 6 people, as buses would be challenging with small children). If you fly to Florence, it might be best (although generally more expensive and at least one layover in Europe is necessary as there are no direct flights from N. America to Florence). If you fly in and out of Rome you should take the train from the Rome airport to Florence as soon as you land (2.5 hours including a likely connection at Rome Termini station, Rome's central station).

4 nights in Sorrento peninsula area (the north side of the Peninsula is called the Sorrento Coast, the south side of the same peninsula is the Amalfi Coast. Sorrento may be logistically better, but you still should use a private driver tour to visit the Amalfi coast, as it might be challenging using the bus with small children. Capri is an easy boat ride from Sorrento (or also from Positano or Amalfi). Traveling from Florence to Sorrento is a 3 hr high speed train from Florence to Naples, then from Naples you take either the commuter train (Circumvesuviana train) for about 5 euro per person or a private driver (about 100 euro).

With 6 people traveling you might also consider a private driver from Rome to Florence, and also from Florence to Naples. It's more convenient for 6 people (but expensive) and not as fast as the train (time of travel by car is at least 50% longer than by high speed train).

If you fly out of Rome, at least the last night must be spent in Rome.

So, what city/area do you recommend adding to Tuscany and Amalfi?

I wouldn't add anything more other than a stay in whichever city you fly in/out of. You will already be losing a good portion of at least 1 day traveling between Tuscany/Amalfi.
Rome is probably going to be your best choice for flights and if you have to do RT then save the Rome stay for last

Fly into Rome- head to Florence that same day- stay 4 or 5 nights wherever you choose. Renting a car for that many people is going to be a bit of an issue- you'll need a van or 2 cars.

After Florence/Tuscany take train to Amalfi area. I'd stay in Sorrento- 4 nights
Train back to Rome 1 or 2 nights
Fly home.

Tuscany and Amalfi will fill 10 days given one day will be travel between the two. Fly Into Florence out of Rome would be my preference as there are more option from Rome to fly home but you could fly out of Naples, I've seen pretty good flights from there as well.

I like Roberto's suggestions.

Traveling with 8 people will be challenging; cutting the number of stops to 1 or 2 would make the trip much less hectic and tiring. I was going to suggest basing in Siena, rather than Florence, but on reflection, Florence may be the better choice. There is so much to see and do there, and you can arrange any number of day trips tailored to fit the wants of your various group members.

Sorrento is a great base, as well, but Amalfi town itself might work better, if it is Amalfi your mom wants to see. Again, there are many interesting day trips from there.