Monk @ 1
Snag stout halfling
Array - 15 14 14 10 10 8 > 10 14 15 8 14 10 > 10 16 16 8 14 10
HP - 11
AC - 15
Attack - main at +5 for 1d8+3 (two-handed staff) and BAA at +5 for 1d4+3 (and the d4 grows with levels)
Edit: can't power-attack with GWM and quarterstaff, requires a heavy weapon, my bad
Honestly, if you have a good tank, the AC isn't particularly terrible to deal with, so you may wish to pick up pam at 4, then gwm, then start buffing dex, unless you're in a low-magic campaign, then buff dex and then wis, and then you'll have your choice of PAM or GWM (both are useful for this build) at 19. However, if you're not looking to progress beyond 10, and you don't care about the math, then grab PAM and GWM (or martial adept) with way of the open hand and just have fun doing weird crap like prone and push and power attack prone targets (actually, that's a solid setup there on big bads since you can prone anything as a open hand monk, grab martial adept, precise strike to make sure you hit (well, +1d6 to roll anyway), bust ki to prone, all the rest of your attacks (and allies' in melee who go before the creature) will be with advantage, and you can do this once per short rest with the superiority die) until they're paste on the ground. But you sacrifice ASIs to grab feats to set that up.
Yeah, I know monk's not super great, but a d8 HD is fine for the majority of campaigns, and plenty of monks have made it to higher levels, so in practical application I don't think he'll be too bad off going monk.
Also, ".. I'm not generally one who builds based on the math. I'm looking at a city watch type halfling, and it felt odd to be a rogue..." from the OP, so if he doesn't care about the math, and is just looking for a nifty concept, I think the monk could work well, since the staff even allows for a modicum of optimization (being able to at-op opponents when they enter reach, an option for bonus action attack, and if he really wanted to, he could use GWM with it if the party finds gauntlets or belt that buffs str that nobody else wants/needs since the staff is versatile, letting him get off some pretty good damage when he two-hands it for a d8+10+prone on an at-op because someone thought they'd just come attack him) down the road.
If I were making the character, he'd totally be a fighter rogue, one of my favorite combos (stupid easy to run, lots of options all the time, consistent damage round to round), but I'm sure he's considered that which is why he's asking for suggestions.
I think a monk town guard totally works too, story wise at least, since you don't have to have ever actually learned from a monastery, that's all fluff. Mechanically you know martial arts and how to defend yourself without armor; you come from a rural area that predominantly relies on livestock and agriculture, being such a ripe fruit for bandits to pluck, the people of the town have learned to wield their farming implements with deadly proficiency, your grandfather taught you how to take every-day items, and even your own body, and use these tools to fight off brigands and thieves. A staff wielding halfling monk totally makes sense for a peasant guard type character, and in the mid-tier can be quite effective in combat as well, while still having good versatility at the low levels. If you go way of the open hand, you're basically the peasant weapon battle master, and can even get some PAM/GWM cheese in there too, thanks to the versatile staff.
But yeah, to be effective at low, mid, and high level tiers in combat and out, fighter rogue is definitely the mechanically better choice, and fluff-wise; you're a crooked guard, you take bribes from the local thieves' guild to look the other way when a crate of wine or a barrel of produce come up missing, and you also get compensation for the information you can provide. Snag the criminal background, I like enforcer specialty for the build since you can then pick up side jobs collecting debts owed by drunk farmers that have bad gambling runs, and you're set. Just go with neutral alignment (you're not chaotic, you're not a monster, you just kind of don't really care about anyone other than you), maybe lawful neutral (there are laws that everyone ELSE is expected to follow, me? I am the law...) if you want to help yourself portray a more self-righteous kind of crooked cop.
