Depending on how you count them there have been about 25 female Doctor Who companions so far, plus a whole bunch of what you might term quasi-companions.  They are all individual and fabulous in their own ways and they have a huge variety of characteristics, performing different functions within different stories.  There is a lot more going on than the old cliché of a girl who passes the test tubes and asks questions.  Let’s see if we can find some connecting themes that run through several different female companions:

• The Grown-up.  The Doctor is a prime example of somebody stuck in a perpetual midlife crisis, generally preferring the company of very young women.  There are exceptions and these tend to be the companions who are pushing more in the direction of being something approach the Doctor’s equal.  They also tend to fall into the “best mate” category and as such have a lot in common with the male companions.  Examples are Barbara, Liz, Romana (but much more in her first incarnation – Lalla Ward ends up wearing a school uniform) and Donna.  It is a minority approach, but it helps with stories that require the Doctor and the companion to solve their own separate mysteries that then come together to make up the larger plot.  Pseudo-companions seem to fit well into this category also, such as Kate Stewart and Vastra.
• The Intelligent Woman.  Obviously all of the companions are brilliant and clever in their own way, but this category of companions is all about the ones who are overtly very clever and that aspect is played on in their stories.  This relieves the Doctor of the burden of doing all the sciency waffling and also solving all the problems himself, freeing him up to have a bit more fun and be as eccentric as possible, or perhaps a different kind of genius.  This is all conveniently forgotten when writers want a companion to be a damsel in distress.  Examples are Barbara, Zoe, Liz, Sarah Jane before she started aging backwards, Romana, Nyssa, Mel, Martha and River.
• The Victim.  These are the ones who have a habit of walking into trouble and getting captured a lot.  They draw the Doctor into a plot because he has to get involved to save his companion, although this ignores his desire to put things right where he sees injustice anyway, so could be argued as unnecessary.  When the captors or torturers are repeatedly male, this becomes a bit of a dodgy repeating theme, most notably with Barbara and Peri.  Other strong examples (most companions fall into this category at least to a minor extent) are Susan, Katarina, Victoria and Jo.
• The Reluctant Traveller.  Some companions don’t start out on their travels intentionally, and this tends to define their edgier-than-usual relationship with the Doctor.  Often it oddly ignores how amazing their experiences are and they sometimes get stuck in a grumpy rut, especially Tegan.  Lesser examples, generally minus the grumpiness, are Barbara, Katarina, Polly, Sarah Jane in her first story and Romana in her first incarnation.  Dodo fails to qualify despite being a kidnap victim like Barbara, because she is magnificently unbothered by that fact.
• The Girl from the Future.  You would think this would make a big difference but generally it doesn’t, beyond the opportunity for the occasional line of dialogue that references it.  Vicki is treated by the writers much the same as a contemporary companion.  For Zoe and Nyssa it is simply a way of placing them in the Intelligent Woman bracket, as if a contemporary human one of those is somehow unlikely.
• The Girl from the Past.  This is a rarity because of the assumption that one of these cannot function successfully as audience identification.  Which is of course nonsense.  Examples are Katarina, Victoria, and the version of Clara we see in The Snowmen.
• The Warrior.  This is a rarity because a companion with tendencies to physical violence should not on paper be the kind of person the Doctor wants to travel with him, but it is something Doctor Who has tried occasionally, mainly for the sake of doing something different.  The first was Sara Kingdom, who kills her own brother before joining the Doctor by necessity.  This is a way of showing what a desperate situation the Doctor is in.  Leela is the most obvious example of the warrior companion, and he spends a lot of time with her trying to civilize the savage, which is all quite imperialistic and rotten, especially as she is at her most compelling as a companion when she is being herself.  Also verging on this category due to her fondness for explosives is Ace.
• The Alien.  This is never as significant as you might expect, because there is nothing really to distinguish the alien companions as aliens.  They always look human and all it does is allows for a slightly more interesting way (or an excuse) to do a particular type of companion, so we have Leela who is a way of doing Warrior + Girl from the Past, Nyssa who is a way of doing Intelligent Woman + Girl from the Future and Romana who is a whole mix of stuff, but most importantly holds up a mirror to the character of the Doctor.
• The Young Contemporary Earth Girl.  Believe it or not, it takes quite a long time for Doctor Who to try this (Dodo and Polly) and then it doesn’t happen again for ages (the Pertwee era companions) and is surprisingly rare in the classic series.  It has become the default ever since Doctor Who returned in 2005.  The obvious reason is audience identification, but it also opens up the companion role to the most interesting way to explore a character arc: a normal young woman with nothing to give her any particular advantage (e.g. future knowledge, super brains etc) learning how to be completely brilliant by travelling with the Doctor.  Examples include Dodo, Polly, Jo, Sarah Jane, and then basically everyone from Peri onwards: Mel, Ace, Rose, Martha, Amy, Clara and Bill.
• The Failed Romance.  Because it always has to be a failure or you collapse the whole narrative of Doctor Who.  The classic series only hinted at this, with Jo chosing for herself a younger version of the Doctor, and Romana having a sort of hinted romance in Paris with the Doctor.  Plus Sarah Jane got ret-conned as the “ex”.  Almost every new series companion, plus Grace, has been defined in terms of some kind of romantic attraction to the Doctor.  Rose was in love with him and it was sort-of reciprocated.  Martha suffered unreciprocated love.  Amy tried to bed him.  There were hints bubbling under the surface with both Donna and Clara, who both get misunderstood from time to time as the Doctor’s girlfriend, and of course Clara just manages to stop herself confessing her attraction to the Doctor in The Time of the Doctor.  The Doctor is clearly such a catch that the only way to stop this cycle was to have a lesbian companion, which is a bit crass when you stop to think about it.
• The Best Mate.   And this is the one that tends to work best.  Vicki pioneered it, as the companion whose influence taught the Doctor how to be the Doctor.  It doesn’t really crop up strongly again until the ultimate best mate female companion, Sarah Jane, but you could also consider several other companions as a reasonable fit: I would say Zoe if it weren’t for the Second Doctor’s much stronger bromance with Jamie; Jo, but she is really more under the Doctor’s wing; Romana, except there is a hint of romance; Peri (eventually); Mel, Ace (definitely), Donna and Bill.

What this all boils down to is a lot of different ways of exploring the Doctor’s character through somebody else’s eyes.  He can be many different things: kidnapper, grandfather, old authority figure, best friend, scientist, wizard, maverick, love interest, professor, messianic figure, but ultimately an amazing friend who never lets the girl down… mostly.   RP