According to Wikipedia and this answer, the solfège syllables for 2 and ♭2 are re and ra, respectively, while the solfège syllables for 6 and ♭6 are la and le, respectively. Is there a reason for this apparent inconsistency?
That's because the solfege syllables for the non-chromatic notes (Do, Re, Mi, Fa, Sol, La, Si/Ti) were first. They were thoroughly historically anchored in music theory, long before someone thought about adding chromatically altered versions of them. Because most vowels were already used it was very difficult to invent a system 'on top' of the already known syllables that changes vowels consistently.
There's no other reason.
The diatonic notes, of course, came first. Sometime after that, the chromatic notes were described by the system. (It's important that we use Ti instead of Si, as you'll soon note.)
To sharpen notes, the vowel sound in the syllable was changed to i (rhymes with tree), as in Di, Ri, Fi, Si, Li, etc. To flatten notes, the vowel sound was changed to e (rhymes with day), as in Me, Se, Le, etc.
Sharp notes were easy; any note that could be sharpened worked just fine, as Mi and Ti already have the I and they can't really be sharp.
But FLAT notes pose a problem. The note "Re" already has the "e", and we often flatten the second. So, they decided to change the flat supertonic syllable to "Ra".
Chromatic scales with sharps and flats:
Do, Di, Re, Ri, Mi, Fa, Fi, So, Si, La, Li, Ti, Do
Do, Ra, Re, Me, Mi, Fa, Se, So, Le, La, Te, Ti, Do
It is exactly like Tim says.
I’m used to sing la flat = lu.
Finally the spelling of the altered syllables doesn’t matter. You could also tell the flattened re and la: ru and lu.
Important is the association of the vocals and the function with the leading tone.
While mi and ti with the vowel i are up leading tones, analogical you can spell all up altered names adding an i:
Di, Ri, Fi, Si, Li
and all down leading tones adding a or u:
Ra, Ma, Sa, Lu, Ta or Ru, Mu, Su, Lu, Tu
The spelling of the altered tones differs from country to country. But this won’t cause any communication problems.
Solfege syllables were derived from a hymn to St. John. The first six lines of the hymn had begun with a different syllable:
Ut, Resonare, Mira, Famuli, Solve, Labii, Sancte | ohannes
Eventually, the Ut and Si were changed to Do and Ti,and this made:
Do, Re, Mi, Fa, Sol, La, Ti, Do
The above diatonic notes were theorized upon first, and for the second there came the chromatic notes.
The vowel sound was changed to i to sharpen notes:
Do, Re, Mi, Fa, Sol, La, Ti, Do -> Di, Ri, Fa, Fi, Si, Li, Do, Di
And to flatten notes, the vowel sound was changed to e. However, we come to a problem: the Re already has the e. So in this case, the vowel was changed to a. In fact, we get:
Do, Re, Mi, Fa, Sol, La, Ti, Do -> Ti, Ra, Me, Mi, Se, Le, Te, Ti
In fact, it may seem strange at first that Ra is lower than Re while La is higher than Le, but once we know the reason above, it will make sense.