If I use Animate Land to turn an opponent’s land into a creature and then enchant it with Darksteel Mutation, could I then use Eradicate to exile all of their lands with that name?
Yes, that's completely doable. Note that Eradicate looks for cards with the same name, not creature cards with the same name. Snow-Covered Plains, having been turned into a creature, is still named "Snow-Covered Plains" and so are all other copies of that card in its owner's library.
EDIT: Darksteel Mutation isn't necessary here unless the land happens to have, for example, protection from black somehow; it doesn't matter that the creature is a land as well, and usually land creatures via Animate Land will be colorless and therefore nonblack.
Eradicate says you search the stated zones for "cards", not necessarily creature cards, with the same name as the targeted "nonblack creature", even if that permanent normally isn't a nonblack creature, so this works as you intend. See also this thread.
EDIT: Darksteel Mutation isn't necessary here unless the land happens to have, for example, protection from black somehow; it doesn't matter that the creature is a land as well, and usually land creatures via Animate Land will be colorless and therefore nonblack.
I̟̥͍̠ͅn̩͉̣͍̬͚ͅ ̬̬͖t̯̹̞̺͖͓̯̤h̘͍̬e͙̯͈̖̼̮ ̭̬f̺̲̲̪i͙͉̟̩̰r̪̝͚͈̝̥͍̝̲s̼̻͇̘̳͔ͅt̲̺̳̗̜̪̙ ̳̺̥̻͚̗ͅm̜̜̟̰͈͓͎͇o̝̖̮̝͇m̯̻̞̼̫̗͓̤e̩̯̬̮̩n͎̱̪̲̹͖t͇̖s̰̮ͅ,̤̲͙̻̭̻̯̹̰ ̖t̫̙̺̯͖͚̯ͅh͙̯̦̳̗̰̟e͖̪͉̼̯ ̪͕g̞̣͔a̗̦t̬̬͓͙̫̖̭̻e̩̻̯ ̜̖̦̖̤̭͙̬t̞̹̥̪͎͉ͅo͕͚͍͇̲͇͓̺ ̭̬͙͈̣̻t͈͍͙͓̫̖͙̩h̪̬̖̙e̗͈ ̗̬̟̞̺̤͉̯ͅa̦̯͚̙̜̮f͉͙̲̣̞̼t̪̤̞̣͚e̲͉̳̥r͇̪̙͚͓l̥̞̞͎̹̯̹ͅi͓̬f̮̥̬̞͈ͅe͎ ̟̩̤̳̠̯̩̯o̮̘̲p̟͚̣̞͉͓e͍̩̣n͔̼͕͚̜e̬̱d̼̘͎̖̹͍̮̠,͖̺̭̱̮ ̣̲͖̬̪̭̥a̪͚n̟̲̝̤̤̞̗d̘̱̗͇̮͕̳͕͔ ͖̞͉͎t̹̙͎h̰̱͉̗e̪̞̱̝̹̩ͅ ̠̱̩̭̦p̯̙e͓o̳͚̰̯̺̱̰͔̘p̬͎̱̣̼̩͇l̗̟̖͚̠e̱͉͔̱̦̬̟̙ ̖͚̪͔̼̦w̺̖̤̱e͖̗̻̦͓̖̘̜r̭̥e͔̹̫̱͕̦̰͕ ̗͔̠p̠̗͍͍̱̳̠r̰͔͎̰o͉̥͓̰͚̥s̟͚̹̱͔̣t͉̙̳̖͖̪̮r̥̘̥͙̹a͉̟̫̟̳̠̟̭t͈̜̰͈͎e̞̣̭̲̬ ͚̗̯̟͙i͍͖̰̘̦͖͉ṇ̮̻̯̦̲̩͍ ̦̮͚̫̤t͉͖̫͕ͅͅh͙̮̻̘̣̮̼e͕̺ ͙l͕̠͎̰̥i̲͓͉̲g̫̳̟͈͇̖h̠̦̖t͓̯͎̗ ̳̪̘̟̙̩̦o̫̲f̙͔̰̙̠ ̹̪̗͇̯t͖̼̼͉͖̬h̹͇̩e͚̖̺̤͉̹͕̪ ͚͓̭̝̺G͎̗̯̩o̫̯̮̟̮̳̘d̜̲͙̠-̩̳̯̲̗̜P̹̘̥͉̝h͍͈̗̖̝ͅa͍̗̮̼̗r̜̖͇̙̺a̭̺͔̞̳͈o̪̣͓̯̬͙̯̰̗h̖̦͈̥̯͔.͇̣̙̝