Personally, I think Harry's not going to make it past year seven in canon... so...
Anyway, even then, if you examine the text carefully, Harry's pretty much not very sociable. The only people he can reasonably call friends are not that many and he isn't very close to them. His buddy, Ron, is a git that gets jealous of small things and is ironically, the most well written character in the series.
Almost everyone around him seems to be a flat caricature, so its difficult to get a handle on them. Yes, even Ron and Hermione are caricatures, Hermione especially so.
Bellatrix Lestrange-Black - Based on canon, no. She's a Death-Eater and pretty insane as well.
Fluer Delacour - I wouldn't mind. Again, she's getting married to Bill. I always thought she would make a good sisterly figure.
Gabrielle Delacour - Too little information to go by. You can start with the Hero Worship part, but very little to work with. Seems to be a popular alternate ship since it would be easier for the user to write her.
Ginny Weasley - Frankly, I have never understood the rationale behind Harry/Ginny. For 4 books, we had little to zero information about Ginny, but she was always a favorite to become paired with Harry largely because Sugar Quill said so. In book 5, she actually had more screentime as a chaser, and I thought it was good to see one more character get developed. In book 6... I try my best not to invoke book 6 as it is the nadir of Harry Potter series.
I blame the One Big Happy Weasley Family syndrome. Harry goes with Ginny and Ron goes with Hermione. Everyone ends up happy.
I can't recommend any fanfic that actually gives this pairing justice. Most of it is so very cliche, even worse than Harry/Hermione. And since it has been around so long, most authors simply just jump in and say that one day, Harry noticed how gorgeous Ginny has become and wants to jump her bones...
Hermione Granger - I think this is possible, despite book 6. She's always concerned about Harry and she's a much better and loyal friend than Ron. Harry shows the same amount of concern for her, which is saying something considering he's an insular kind of chap. So there's the basis of taking it beyond mere friendship.
It doesn't hurt that some of the best fanfics I've read are centered around the two of them. I recommend the Paradise series (featuring adult!Trio) and Year of Rebellion.
However, I always get the feeling people are working with Emma Watson as Hermione and Daniel Radcliffe as Harry. They do look good together eh... but in canon, Hermione is no gorgeous babe and Harry is a tad too scrawny for his own good... was hoping people take notice of this...
Lily Potter-Evans - (Arcs an eyebrow). This brings Oedipius to a new level eh...
Luna Lovegood - I've read some good ones. Luna has alot in common with Harry, so I see this as perfectly reasonable. Luna's Year seems to be one. Year of Rebellion as a date scene between Harry and Luna and I like the Luna Full Pensieve wrote.
Minerva McGonagall - I think there's a reason why no one voted for her.
Nymphadora Tonks - Erm, again, she's with Lupin. She's seem to be the mischievous older sister type rather than girlfriend. Then again, I've read some that are quite good... then to be a bit smuty though.
Narcissa Malfoy-Black - read Bellatrix's entry.
Pansy Parkinson - There's a good one at fanficauthors.net... Is it by Jeconais? Not too sure...
Rowena Ravenclaw -... Eh, this is grasping at straws...
Some other options -
Cho Chang - I've actually read several good ones. Most of them revolve around Cho being actually a normal human rather than a weepy caricature shown in book 5.
The other girls - I've read alot of other stuff... but mostly require Harry to actually socialize outside of his established group. Not really feasible, but hey, if its well written, its okay by me.