If I change the name of one of my computers, does anyone know what happens to the credit accumulated under the original name? Do they stay with the original name (same computer) or do they follow the computer regardless of the name?

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Nothing happens to the credit accrued by your computer because SETI doesn't use your computer name(s) but the unique computer identity number.
(Only you can see your computer name(s) in your list of computers, the rest of us just see that unique number)

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Bob Smith
Member of Seti PIPPS (Pluto is a Planet Protest Society)
Somewhere in the (un)known Universe?

More specifically, credits etc. follow the 'HostID' number which is held within the whole collection of data stored in BOINC's data folder and subfolders (stored on your computer). You can move that data folder to a different computer with a different name, and the statistics will follow the number, not the name. Alternatively, you can change the number - something which happens from time to time by mistake - and start a new set of statistics under the same name.

Just don't try to cheat the system by using the same HostID number on two different machines at the same time. BOINC will detect this, and change one (or possibly both) numbers so the separate machines are kept separate.