This is a tough one because with interest rates how they are, it appears that by paying down the mortgage faster, it saves you money in the long run. Not knowing the details of your situation, I'm going to make some assumptions. It sounds like you don't have significant savings right now (because you are borrowing $5000 for the renovation). Depending on where that $5000 comes from, you are going to be paying more or less interest on that borrowed money than your mortgage interest, but definitely more than you would earn on an interest-bearing savings account. So, although it seems that the less than 1% you earn for savings isn't significant, what that savings would actually be doing is preventing you from *paying* more in interest when you need extra cash. If you think of it that way, perhaps it will help you decide. Sometimes savings isn't about how much interest you earn on it... sometimes it's about the money it saves you from having to get a loan with interest that you will pay. You may not have earned much on $5000 in a savings account, but it would have saved you a few percentage points on what you now have to borrow because you don't have it liquid. (I hope that rambling explanation made some sense.)
IMO, before paying off a mortgage early, especially if it's not your forever home, you should look into making sure you are maxing your contributions at work (up to match, even though you're not fully vested yet), that you have an 8-month emergency fund (8 months of expenses in a savings account) and that you are contributing to a Roth IRA. I think too little emphasis is put on retirement these days and it's the one thing that, under any circumstances, *cannot* be financed. You cannot borrow to retire. College, housing, cars, etc. can be financed, though. Looking 20, 30 (maybe even 40) years down the road is just as important as the short-term financial goals you set for yourself. It sounds like you have the short term goals established, but I'm not sure (based on your post) that you've looked long-term. When I say long-term, I mean at least 20 years in the future.
All that being said, I don't think it has to be all or nothing - savings OR mortgage pre-payment. You can save a larger portion and pay down your mortgage a little, then once you have what you want in savings, flip-flop the portions (or even reallocate to do other types of investing such as a 529 or taxable investing). You said that you "hope" that extra payments will build equity. Does that mean that where you live, the housing market has not reached its low? Most haven't. You definitely don't want to put all of your eggs into one basket, having no savings and all of your money tied up in your house, then not be able to tap into it with a home equity loan. What if you don't build enough equity and when you need a sum of money (such as the $5000), you can't get it because you're upside down? The past few years have shown us that we have no control over that. The housing market can take a sudden turn for the (even) worse and all that money put into the mortgage could be lost. Building equity is no longer a guarantee. Something to think about.
I find that an extra payment calculator can be beneficial to know just how much you are saving when you make extra payments or larger payments than usual on your mortgage. You can get your numbers then determine if it's worth it. In the long run, it probably is, but I think there are other investment avenues you would want to explore before doing so, such as an emergency fund, sinking funds for home repair, appliance replacement, vehicle replacement, Roth IRA, college funds, etc. If it was your forever home, I would have a different opinion. Each financial situation is unique, so none of this may apply to you or not be right for your situation. We are probably older than you (well, you're TTC, so I know we're older than you), so there is some personal experience I'm tapping into and advice given to us by professionals in the past. Good luck with your decision.