So what does this seemingly innocuous word mean? From the couple grammar books at my disposal, here's what I put together for the Modalpartikel 'eben':
eben/halt : emphasizes a known or inescapable fact. Unabänderliche Konsequenz, Resignation. "Max ist eben/halt Saras Freund". "Die letzte U-Bahn für heute ist vor 5 Minuten abgefahren. Dann müssen wir eben zu Fuss gehen". "Warum willst du denn nicht? Ich will halt nicht"
Does this have anything to do with äbä? Can somebody provide a couple examples please?
Aber = but; with various dialect versions, äbä not being one however unless you're very lonely living in the Walliser Alps. An ä is simply ae. Ebenhalt or eben is the emphasis as your example indicates.
"Dann müssen wir eben zu Fuss gehen" = Then we'll just have to go by foot" The eben is the just part in this phrase, it emphasizes that plan A has clearly not worked out, so we'll just have to go on foot. That's why English is a superior language.
I have a german speaking child who definitely uses it as 'but'...
My toddler has come home from swiss-german preschool, and starts every sentence with something similar to 'alle' - which we assume is something like an adult telling the kids 'everyone....we are going to bla bla bla'...
If in doubt, just throw 'em all together: " äbä stimmt genau doch äbä".
Swiss German speakers IMO like to use several words rather than one. See, for example, people saying goodbye: "Tchueesss aedieu Ciao Ciao aufwiederlueegaa bis nacher Tchueessli..."
Germans, on the other hand, are a cold bunch: " Auf Wiedersehen".