Some players do opt to stay for the senior year but it is a big gamble and I think that most players will take the money and the draft pick and go.
One could very easily argue that it would be a bigger gamble to pass on finishing your degree over what amounts to a max of $50,000 cash (post agent fee and taxes) and a minor league salary (around $2,000 per month for 6 months out of the year). If you don't make it to the bigs, your professional career is over at 26, you've made less than just about any UConn graduate would make by that point, and you don't have a degree.
The odds of McDowell and Siena doing so poorly in college their senior year that they don't get drafted next year are asymptotic to 0%. If that does occur, odds are they never had a chance at making it to the majors anyways. You hear success stories of guys making it to the show after being drafted in rounds 11-40, but only ~10% of players from rounds 11-40 ever even make their MLB debut, and a good amount of those only get a cup of coffee and are never heard from again.
I understand that your son is on the team so you probably have a better insight into what individual players might be thinking, but I can assure you that the people on this board know what they are talking about when it comes to all things college sports and in many ways can be more objective since they do not have family on the team. I don't know who your son is, but if he is ever given the choice between a $50-100k bonus or returning for his senior year, I would strongly suggest you consult Coach Penders and listen to the advice he gives you. I understand some families are in worse financial situations than others so they may be tempted to take the money up front, but I would bet that in the long run a degree from UConn will be more valuable than the average career earnings of a player drafted after the 10th round.
I hope I don't sound condescending in any way as that's not my intention at all. I fully believe Max and Vinny will have every opportunity to make it to the majors. Hell as a Mets fan I would love to see Siena and Mazzilli on the team at the same time one day. I understand that playing in the majors is the dream of every college player, but these are life decisions and you have to look at them objectively. As the adults in the situation, it is the responsibility of the parents and coaching staff to guide the player to the best long run decision even if that means telling him to put his dream (and likely the parent's dream) on hold for one more year. Now it may very well be the right decision for a given player to sign for his $50k, and if after weighing all the factors that's the conclusion he and his family reach then by all means he should sign. But I would urge them to make sure they look at things realistically before reaching that decision.