You can use the memory write {address} lldb command to overwrite the memory and change the string value. I managed to do it one address at a time, but it seems like memory write is capable of doing it one go.
(lldb) help memory write
Write to the memory of the process being debugged.
Syntax: memory write <cmd-options> <address> <value> [<value> [...]]
Command Options Usage:
memory write [-f <format>] [-s <byte-size>] <address> <value> [<value> [...]]
memory write -i <filename> [-s <byte-size>] [-o <offset>] <address> <value> [<value> [...]]
-f <format> ( --format <format> )
Specify a format to be used for display.
-i <filename> ( --infile <filename> )
Write memory using the contents of a file.
-o <offset> ( --offset <offset> )
Start writing bytes from an offset within the input file.
-s <byte-size> ( --size <byte-size> )
The size in bytes to use when displaying with the selected format.
This command takes options and free-form arguments. If your arguments
resemble option specifiers (i.e., they start with a - or --), you must use
' -- ' between the end of the command options and the beginning of the
arguments.
Here's an example (hopefully someone with more understanding of the lldb and the internals of Swift can provide a better method):

This shows overwriting the memory one byte at a time. po "Tom".dataUsingEncoding(NSUTF8StringEncoding)! gets the hex representation, which is used to step through and overwrite the memory of object.name. I'm sure there's an easier way to do it (in one command), but I couldn't figure out the correct parameter values to pull it off.
Object, why do not you initialize a newObjectrather than change the name of it?