I was actually thinking about this for a while and I think I would go with Salandit. Probably an odd choice but I like lizards/salamanders and I loved Salandit from the moment it was revealed. I think it has one of the best designs in the entire Pokemon series. So it would be awesome to have one as a "pet" and take it everywhere with me. Salandit is a small Pokemon so it would be super convenient to keep it outside of its Pokeball all the time. ☺
I wanted to avoid the obvious answers like starters and "dogs"/"cats" of the Pokemon world like Rockruff,Growlithe,Eevee,Skitty,...
I if was to go with one of these more obvious ones it would probably be either Totodile or Growlithe.