I switched to fifths tuning some years ago, and never could get along with roundwound strings. Plus I mostly play DB, which is probably a big part of the not liking roundwounds. So, unless I want to have inconsistent tensions, that means d'Addario Chromes or octave4plus custom flatwounds are the only real options, and the Chromes only come in one length. So, that pretty much narrows it down to 4-strings (tried tuning a 5 FCGDA and I really don't need the extra low notes) with a 34"-35" scale length, though I did get custom strings for a hollowbody with a couple of extra inches of string behind the bridge. In theory a shorter scale with a few inches behind the bridge that would take normal-length strings would work, but with the low C and being used to DB, I don't really want a shorter scale either.
I also play lefty, but strung righty, and I've mostly played right-handed basses over the years. I've owned two proper lefties and didn't really see much benefit (even though one was a very nice 70s BC Rich, back when they weren't so expensive). But for some reason I like larger bodies, and a Thunderbird upside down feels small because there's not much under the forearm. My four BGs right now are a big deep hollowbody around 16.5" wide with a single right-handed cutaway, a big thin chambered body around 15.5" with a similar cutaway, a Flying V which is symmetrical, and a Killer Simmony which is kinda shaped like a Ricky but upside-down and then I play it lefty so the shape is right side up again... it's just funny.
So, scale length and body shape are a big deal. So's number of strings and frets - I've owned a couple of fretless basses and don't really feel like I need one. Another factor that isn't in the poll: I prefer bigger necks, but J-width is OK, and so is somewhat narrow string spacing at the bridge (which both my hollowbodies have), so those won't keep me from trying a bass. Neither will color or finish - in theory I like things bright and shiny and eyecatching, but the big hollowbody is a satin brown sunburst... Electronics, well, all my basses are passive, but if an active one looks really cool I can live with a battery and EQ knobs which I'll probably just set one way and keep forever. I don't really need a bridge pickup, either.
Weight doesn't matter much either, I was totally comfortable with an 11+ pound bass on a normal-width strap for three-hour rehearsals, though moved on from that bass because I had no need for the fifth string. Fingerboard material, headstock shape I've never really thought about, though they do need to match the overall look of the bass. Brand won't keep me from trying a bass; the Flying V is homemade and unbranded after all.
Price, well, OK, probably, everybody has their limits... though I'd try a $4000 bass and hope I can someday find one used for $1500, heh. That's pretty much the ESP hollowbody I have. And I've played a bass I bought for 40 British pounds on a gig.
Sorry for the novel-length post, heh. I started thinking and typing and just kinda kept going.