I know Jock Scot suggested me but as I never stay in any Edinburgh hotels I am probably the last person to advise you. What I would say is that you might be best picking up a hire car at Edinburgh airport then driving to a hotel for the night before setting off on your travels. Arnold Clark have a depot there and I know them to be a highly reputable firm if you are looking for suggestions. There will be other big name rental companies there too.
I would not recommend driving into Edinburgh. It is a nightmare at the moment in the centre with roads closed due to tram works and even for locals finding your way around is a problem. It is not dangerous so don't be worried about that, but the route your map tells you is likely to be blocked and finding an alternative is hell. believe me.
I take it you are planning to "do Edinburgh" at the end of your visit so you would be best heading off on your travels and find a hotel en-route. There are plenty of Travelodges around and try TripAdvisor for plenty of helpful advice.