Splinter can target any artifact, even if it normally isn't one (for example, if it's an artifact only due to Liquimetal Coating's effect) (C.R. 114.1a; see also C.R. 205.2b). Splinter, however, doesn't destroy that artifact, but rather exiles it (see the updated Oracle text [C.R. 108.1]).
So could you theoretically turn a basic land into an artifact, then cast splinter and remove all lands with the same name (ie mountains)?
Yes. If the basic land's an artifact at the time of targeting and of resolution, then even though other land cards in hand/library/'yard aren't artifacts, they still share a name and get exiled.
EDIT (Jan. 3): Add rule citation.
Yes. If the basic land's an artifact at the time of targeting and of resolution, then even though other land cards in hand/library/'yard aren't artifacts, they still share a name and get exiled.
I̟̥͍̠ͅn̩͉̣͍̬͚ͅ ̬̬͖t̯̹̞̺͖͓̯̤h̘͍̬e͙̯͈̖̼̮ ̭̬f̺̲̲̪i͙͉̟̩̰r̪̝͚͈̝̥͍̝̲s̼̻͇̘̳͔ͅt̲̺̳̗̜̪̙ ̳̺̥̻͚̗ͅm̜̜̟̰͈͓͎͇o̝̖̮̝͇m̯̻̞̼̫̗͓̤e̩̯̬̮̩n͎̱̪̲̹͖t͇̖s̰̮ͅ,̤̲͙̻̭̻̯̹̰ ̖t̫̙̺̯͖͚̯ͅh͙̯̦̳̗̰̟e͖̪͉̼̯ ̪͕g̞̣͔a̗̦t̬̬͓͙̫̖̭̻e̩̻̯ ̜̖̦̖̤̭͙̬t̞̹̥̪͎͉ͅo͕͚͍͇̲͇͓̺ ̭̬͙͈̣̻t͈͍͙͓̫̖͙̩h̪̬̖̙e̗͈ ̗̬̟̞̺̤͉̯ͅa̦̯͚̙̜̮f͉͙̲̣̞̼t̪̤̞̣͚e̲͉̳̥r͇̪̙͚͓l̥̞̞͎̹̯̹ͅi͓̬f̮̥̬̞͈ͅe͎ ̟̩̤̳̠̯̩̯o̮̘̲p̟͚̣̞͉͓e͍̩̣n͔̼͕͚̜e̬̱d̼̘͎̖̹͍̮̠,͖̺̭̱̮ ̣̲͖̬̪̭̥a̪͚n̟̲̝̤̤̞̗d̘̱̗͇̮͕̳͕͔ ͖̞͉͎t̹̙͎h̰̱͉̗e̪̞̱̝̹̩ͅ ̠̱̩̭̦p̯̙e͓o̳͚̰̯̺̱̰͔̘p̬͎̱̣̼̩͇l̗̟̖͚̠e̱͉͔̱̦̬̟̙ ̖͚̪͔̼̦w̺̖̤̱e͖̗̻̦͓̖̘̜r̭̥e͔̹̫̱͕̦̰͕ ̗͔̠p̠̗͍͍̱̳̠r̰͔͎̰o͉̥͓̰͚̥s̟͚̹̱͔̣t͉̙̳̖͖̪̮r̥̘̥͙̹a͉̟̫̟̳̠̟̭t͈̜̰͈͎e̞̣̭̲̬ ͚̗̯̟͙i͍͖̰̘̦͖͉ṇ̮̻̯̦̲̩͍ ̦̮͚̫̤t͉͖̫͕ͅͅh͙̮̻̘̣̮̼e͕̺ ͙l͕̠͎̰̥i̲͓͉̲g̫̳̟͈͇̖h̠̦̖t͓̯͎̗ ̳̪̘̟̙̩̦o̫̲f̙͔̰̙̠ ̹̪̗͇̯t͖̼̼͉͖̬h̹͇̩e͚̖̺̤͉̹͕̪ ͚͓̭̝̺G͎̗̯̩o̫̯̮̟̮̳̘d̜̲͙̠-̩̳̯̲̗̜P̹̘̥͉̝h͍͈̗̖̝ͅa͍̗̮̼̗r̜̖͇̙̺a̭̺͔̞̳͈o̪̣͓̯̬͙̯̰̗h̖̦͈̥̯͔.͇̣̙̝