Is there a way to write the names "Hugin" and "Muninn" in Norse runes? Are there differences in the meaning of each ("memory" and "thought")?
Of course you can write their names in runes; runes is simply another alphabet (or rather: several alphabets). Here's what it would look like, using the Younger futhark:
(Note that while the names in Old Norse would more properly be "Huginn" and "Muninn", consonants are not duplicated in the same word, sometimes not even across words).
If you want them together, it would be
ᚼᚢᚴᛁᚾ:ᚬᚴ:ᛘᚢᚾᛁᚾ Hugin ok Munin
(There were also other separators between the words that could be used, but the colon is easiest to reproduce; it is also used in the runic manuscript of the Scanian law).
As for if the words would also carry a common meaning, from what I gather, they would likely. I found a dictionary from the University of Copenhagen with words found in Old norse prose. For "huginn", there is an entry, but no translation given (but there is one for "hugi": thought, mind, (sincere) feeling". There is no entry for "muninn", but I found it in a free search, so it seems that there is at least an homograph. This also means that since they are inflected forms, "Hugin and Munin" does not necessary exactly translate to "thought" and "memory"; the listing for "huginn" marks it as an adjective. Also note that I believe that the standard word for "memory" would be "minni", at least that is the word that has survived into modern nordic languages.
Lars Magnar Enoksen, Runor.