I call this PƏÆTK-BEADG-CF system (circle of fourths/counter-clockwise) and GDAEB-KTÆƏP-FC system (circle of fifths/clockwise) (I just use dashes to group those patterns into chunks). The note names are based on the circle of fifths/fourths - and the occuring pattern: BEADG (circle of fourth/counter-clockwise pattern) - and the counterpart flat note names influenced by IPA/International Phonetic Alphabet.
How did I come to this solution?
As you may know, the notes in circle of fourths in flats are (starting from Bb):
Bb Eb Ab Db Gb B E A D G C F
or the same in sharps:
A# D# G# C# F# B E A D G C F
Immediately one can see the common ground between the sequences of flat notes and natural notes:
Bb Eb Ab Db Gb
B E A D G
So what I've done is I have taken those flat notes and assigned them new close phonemes (influenced by IPA/International Phonetic Alphabet):
Bb = P (voiceless counterpart letter of B)
Eb = Ə (pronounced [ə], as in words "a" [ə] or "the" [ðə], particularly derived from English word "the")
Ab = Æ (pronounced [æ], as in words "and" [ænd] or "man" [mæn])
Db = T (voiceless counterpart letter of D)
Gb = K (voiceless counterpart letter of G)
So the note letters starting from C, are:
C T D Ə E F K G Æ A P B
The circle of fourths (going counter-clockwise in the circle of fifths) are in my system (starting from P):
P Ə Æ T K B E A D G C F (PƏÆTK-BEADG-CF)
And the circle of fifths (going clockwise in the circle of fifths) are in my system (starting from G):
G D A E B K T Æ Ə P F C (GDAEB-KTÆƏP-FC)
The system helps to immediately to find the notes in a guitar.
Circle of 4ths. In a standard E tuned guitar, one can easily find those patterns (using my PƏÆTK-BEADG-CF notation system). Counting the notes per fret, starting from the 6th string, just remember to step one fret to left at the 2nd string:
E| F K G Æ A P B [C] T D Ə E F K G Æ A P [B] C T D Ə E
B| C T D Ə E F K [G] Æ A P B C T D Ə E F [K] G Æ A P B
G| Æ A P B C T [D] Ə E F K G Æ A P B C [T] D Ə E F K G
D| Ə E F K G Æ [A] P B C T D Ə E F K G [Æ] A P B C T D
A| P B C T D Ə [E] F K G Æ A P B C T D [Ə] E F K G Æ A
E| F K G Æ A P [B] C T D Ə E F K G Æ A [P] B C T D Ə E
Figuring out only one fret-full of notes. The notes are the same on 1st string and 6th string. Imagine that there is a 7th string. Count one forward from the 1st string/6th string with GDAEB-KTÆƏP-FC and you'll get the correct note for the 2nd string:
E| F K G Æ A P [B] C T D Ə E F K G Æ A P B C T D Ə E
B| C T D Ə E F [K] G Æ A P B C T D Ə E F K G Æ A P B
G| Æ A P B C T [D] Ə E F K G Æ A P B C T D Ə E F K G
D| Ə E F K G Æ [A] P B C T D Ə E F K G Æ A P B C T D
A| P B C T D Ə [E] F K G Æ A P B C T D Ə E F K G Æ A
E| F K G Æ A P [B] C T D Ə E F K G Æ A P B C T D Ə E
B| (K)
Circle of 5ths. Again, count with GDAEB-KTÆƏP-FC, all way up from the 6th string to the 1st string, remember the one fret jump on the 2nd string:
E| F K G Æ A P B C T D Ə E F [K] G Æ A P B C T D Ə E
B| C T D Ə E F K G Æ A P [B] C T D Ə E F K G Æ A P B
G| Æ A P B C T D Ə [E] F K G Æ A P B C T D Ə E F K G
D| Ə E F K G Æ [A] P B C T D Ə E F K G Æ A P B C T D
A| P B C T [D] Ə E F K G Æ A P B C T D Ə E F K G Æ A
E| F K [G] Æ A P B C T D Ə E F K G Æ A P B C T D Ə E
If one deliberately needs to determine whether it's a sharpened or flattened note, one could go with those previous suggestions of adding -eesh (sharp) and -eef (flat) to the the notes (-eesh/-eef only determine whether it's a flattened note or sharpened note, those names, -eesh/-eef, do not themselves flatten or sharpen the notes in this system, as there are already the names T Ə K Æ P for flattened/sharpened/black notes):
C# = Teesh, D# = Əeesh, F# = Keesh, G# = Æeesh, A# = Peesh
Db = Teef, Eb = Əeef, Gb = Keef, Ab = Æeef, Bb = Peef
NOTE: There is also similarities between sharp notes and the natural notes:
A# D# G# C# F#
A D G C F
But I'd rather go with the flat note based system, PƏÆTK-BEADG-CF. BEADG (mnemonic: "bead-G" or "bitchy") or BEADG-CF (mnemonic: "bitchy cipher") is also pronounced more easily and more easily remembered than ADGCF.