While I have heard that Minnie and Friends does sometimes have hours before the park opens, which gets you physically in earlier, it doesn't help you get on rides earlier.
Since the early morning few hours are vitally important for getting many things accomplished (easily, that is), I find it best to have a simple breakfast in the morning, get into the parks and play in Fantasyland before lines build (you cannot believe the difference in the line for Peter Pan 5 minutes after the park opens vs 20 minutes after), get a fast pass or two for later, etc. Get those couple/few hours done with many fun experiences under your belt, and THEN break for either a late breakfast or early early lunch.
So I would have some simple foods on hand, play in the park when it opens (getting there well before opening so you're among the first in the park), then later go to Minnie's for the character breakfast (probably around 10:30 is how I would do it).
But really, and know that we started our adult trips to
Disneyland with one day trips with a little kidlet, I wouldn't take ANY time for a character meal with a one day trip unless it's REALLY important to you. Since I myself was a skittish child who had VERY bad reactions to characters, and we didn't yet know how DS would react, for the trips when he was 17 months old and then one year later, we didn't do character meals on our one day trips. It was during the trip when he was 2 that we realized he's blissfully UNlike me, and LIKES characters so for our trip when he was 3 (big package-deal trip) we did do a character meal. But when you only have one day, unless you really know how a little one is going to react, or really want to do the big meal anyway, I would skip a character meal.
Have you determined if your one day is on a Magic Morning day? That's Tues, Thurs, Sat, and Sun, and if it's a MM when you're not eligible, it means that when you enter the park, Fantasyland and Tomorrowland will already be full of people and have long lines. If you're staying onsite, you're eligible for MM on a MM day, by the way, even if you only have a one day ticket.
Lastly, while DL was the only park we visited on those one-day trips...when we finally made it over to DCA we *regretted* it. Big time. Bug's Land is SO much more little kid friendly. No "scary adventures", no dark and creepy line for Peter Pan, no scary Queen of Hearts, an easier "teacup" sort of ride (Ladybug Boogie)...just a pretty space with music from Toy Story and Bug's Life, with outdoor rides. Our family likes bug's land so much better than fantasyland now...we might not be typical, but you won't know until you go.
But I wouldn't think less of you at all if you do what we did, and just go to DL for your first one-day trip with a 2 year old. Most everyone does that.
Does Minnie's open at the same time relative to park opening every day or does it depend on whether there is MM/EE offered that day? If it's a Wednesday or Friday, for example (non MM/EE) days, and DL opens at 8:00, does Minnie's also open at 8:00? Or does it open at 7:00 (and thus, the "park access unavailable" message? On Thursday, Saturday, Sunday, (assuming a regular park entry at 8:00, EE/MM at 7:00) does Minnie's open at the same time as the EE/MM (7:00) or an hour earlier (6:00 
) or at regular park entry time at 8:00? Or is this all a moot point as it just depends on the season and the day?
