Best advice on how to utilize house

[Hagop Nercessian]

Hey everyone!

Little backstory...my mother currently lives in my childhood home that is totally paid off (father purchased it back in 1985 for like 100k). My father passed away 4 years ago.  The house is in pretty good shape, and is in a very desirable town (A+ schools, the house is valued at 600k+, etc).  There is no shortage of people wanting to move into the town.

My sister and her husband bought a hose, and my wife and I currently rent a apartment so it's just my mom in the house.  It's not a huge house, but a little on the larger side for my mom by herself.  So we have been discussing on and off as a family what would be the best way to proceed with things.  My mom wants to downsize sooner rather than later.  

We all enjoy real estate, and would like to figure out a way to possible start investing.  With that said, we have discussed some options.  But I wanted to ask you all for some other opinions!  We can sell the house (my mother inherited through my fathers will) and use the funds to buy a smaller, cheaper house outright for my mom, and use the leftover funds as down payments on investment properties. (We can most likely get 650k+ for the house and downsize my mom for less than 450k).  Or we can rent out the house and cash out refi and use those funds to buy my mom a smaller house outright.  Ideally, we wouldn't want to get rid of the house since it is in a high demand area.  

What do you all think?  Are we missing a better option?  And the numbers I have provided are ballparks, but I have a pretty good pulse on the values in the town. 

Any advice or ideas are appreciated! Thank you all.