I'm trying to memorise the notes on the fretboard so that I can more easily apply some of the chordal and scale theory I've been learning, but I downtune quite a lot (half-step down, whole step down, and sometimes with a dropped D too). I'm wondering whether in those cases it's conventional to simply override the downtuning and still refer to the Eb as an E, Ab as an A, etc. I realise the dropped D string would need specific attention, but as for the rest of the board, is that something that's done? Or is it common to be able to just offset in your head after a while?
If what you're trying to do is memorize the note names, then it's best to stick with one tuning, and learn the notes in that tuning before moving on to others. Standard tuning is a good choice since it is the "standard". Learning the note names in this tuning will facilitate many other types of learning and communication with others within the guitar curriculum(s) and make it easier to use written notation.
We can call what you're using "standard tuning" even if it is one half step down. In this case you are utilizing a relative tuning where the relationship between the strings is the same as standard tuning. Imagine you don't have a tuner or reference pitch available and just tuned the strings to what you thought was "E" standard tuning. In that case you'd call the low "E" string "E", the next string "A" etc, regardless of the actual pitch sounded.
An important exception to the above: If you happen to be blessed with perfect pitch, then you should tune up to "E" and not "Eb" so that you associate the correct note names with the correct pitches.
Once you have a handle on standard tuning you can learn other tunings as separate entities with offsets from the standard -- if this is even necessary for your style of music and the situations you will encounter with other musicians.
There are cases where there might be good reasons not to take this approach - but - focusing on the learning process - I think it's best not to vary anything when you're tackling a challenging task like learning all the notes on the fingerboard.
Think of the reason you're giving names to notes: communication.
So, it depends who you're communicating with, their expectations of you, and your expectations of them.
When I play soprano ukulele, I think in terms of the guitar fretboard - the intervals between strings are the same as the top four strings of a guitar. So in my head, I play a guitar "D" shape, and my head thinks "D". But in fact, that's a G.
If I'm playing solo, that does no harm, but if I'm duetting with (say) a pianist, though, there will be problems from the start. I say "OK, we open with a D major chord", I strum my D shape, which is actually a G. The pianist plays D, F#, A, and it sounds terrible.
The same happens if you're playing with a bassist who's tuned conventionally, while you're downtuned. You shout "Louie Louie in G", you strum a G shape, which is a Gb chord. The bassist plays an actual G, and it sounds awful.
Of course if the whole band tunes up or down the same amount, then it doesn't matter.
Otherwise, you need to agree how to communicate.
- The easiest thing for your fellow musicians, is for you to communicate using the real pitches. That way, they don't need to know how you've tuned your instrument.
- However, if they're better at mental transposition than you are, it might be OK to shift that load onto them. "OK, guys, I'm playing with a capo on the third fret here, so when I say 'A' I really mean 'C', is that OK?"
- Or you could dodge the issue by talking in relative intervals (after agreeing a key). Instead of "C, F, G", "One, Four, Five".
Orchestras have a similar problem: quite a few brass and woodwind instruments are "transposing instruments", in which the note names don't match standard pitches. Scores written for these instruments are transposed accordingly, and the conductor must remember to use the note names expected by those players, when speaking to them.
Especially if everyone in your ensemble follows suit, you can think of this as playing with a different reference ptich frequency.
Nowadays, most tuning uses a reference pitch of A'=440Hz. You can think of down-tuning your guitar as redefining this reference pitch to be a different, lower, frequency. For example, declaring A'=415Hz is the same as down tuning by one semitone.
As @supercat mentioned, down-tuning can also be thought of as applying a capo on a negative fret. In almost all instances guitar charts involving capo give the chords/notes relative to the capo. In your case, this corresponds to referring to the open 6th string as just "E", even though you've down-tuned relative to concert pitch.
I often play downtuned, and tend to keep thinking of the strings as the standard EADGBE - so you'll play a C chord (for instance), but it'll sound at a lower pitch depending on your tuning.
Obviously, if you play with other people who aren't downtuned then you'll need to get used to transposing the chords/notes you're playing to match them.