Gwilym and I actually live in Beitou! We love it. Been living there for almost four years now.
Quite a few restaurants have opened up near and under Xin Beitou MRT station. All very family friendly.
You can also rent a bike (I've seen many parents with bikes with kid seats attached) and cycle up to Guan Du or even Tamshui from Beitou.
If you like chicken, you should defiantly try "半嶺桶仔雞". Pop that into google and it should give you plenty of reviews and a google location. It's a mountain roasted chicken. My meat eating days are behind me, but I do highly recommend and the veggies they serve are great so I never feel left out.
If you want a nice walk with your family you can also check out 農禪寺 temple (by 奇岩MRT). It's one of the most elegant buddhist temples I have ever been to.
http://gd-park.org.tw 關渡自然公園 Guan Du Nature Park has fun activities for kids. Probably better for 5+ too but! Get to know local birds and do some arts and crafts (nicely airconed building). I often take my dad there for a coffee. Nice walks around the park. After some strong weather it can look a bit in tatters, but lovely on a sunny day. I've seen a lot of cool birds there. If you are lucky, the water buffalos come out!
We should do a Beitou chinesepod meet up!
Not in Beitou, BUT...For poddies with kids 5+, I also highly recommend the Paper Museum (way more fun than it sounds). You can sign up for the tour, make your own paper and kids get a fun and informative talk on how paper is made. http://www.suhopaper.org.tw/en/en_museum.html