There is hardship, pay day next week, reduced working hours, some one is sick, car broke down.

Your equity is reducing your possible returns not increasing your cash flow.If your father has no use for additional income he could leave it where it is and not care if it is impacted in a market shift or he could pull it out and invest in a income fund or a REIT. 

In may of this year I was going to increase the rent but I gave him a chose to pay $50 more per month or if he signed a two year contract I would keep the rent at $850/month.

The fact that housing has increased much more than rents historically indicates that rent appreciation will be greater than property appreciation for a while. 

Rent is at the market so not looking to increase for this term. 

I'm working as an Insights Strategist during the day, putting in a few hours of marketing consulting work on the side during the week, and studying up on real estate investing on the weekends (and whatever time I have left over during the week).Having worked in the finance industry for the last couple years, I became increasingly aware of different types of investing vehicles, but never really considered real estate investing until I moved to Miami this last July.

If the tenant has a lease that expired a year ago, (ie: nobody can produce a current lease), then you just need to draw up a new lease, provide a month's notice of the increase, and if she won't sign or move out after the 30 days, consult a real estate lawyer or file for eviction.

However this would cause your interest rate to increase.  

Also, can't I also just issue a 60 day notice to increase rent by $200 or so.

How does everyone handle rent increases with new home purchase, with existing tenants?