It is a problem that Lost Mines of Phandelver is designed for 3-5 player characters (PCs), plus a DM. I'm playing Phandelver at the moment with a group. One session only two players turned up, and I discovered how deadly this could be with no modification. With one PC and no modification, you would likely not last past the first encounter. As far as I can see it, you have options:
1) Tone down Lost Mines of Phandelver
By 'tone down' I mean reduce the level of the encounters. This will be easier near the beginning (where you could just reduce the number of goblins), and more difficult as the game progresses.
I actually did this with the session in question - they were supposed to encounter 6 hobgoblins (rolling on the random encounter table at that point in the game), but I cut it down to two.
With bigger monsters it is more difficult, though you could 'wound' them, reducing their HP (hit points) and possibly some of their special abilites, and XP (experience points) commesurately, but that can get complicated and possibly lack verisimilitude, so there is always option 2).
I didn't do this to one monster (a grick) and both characters ended the session unconscious. Oops.
2) Play Phandelver, but have your girlfriend control two or more characters, and/or bolster the party with an NPC or two
In this solution, you can still play Lost Mines of Phandelver, but you could split the party between you and your girlfriend, with your girlfriend controlling two PCs and you controlling another two (or three if you want to use all five) as NPCs (non player characters - controlled by the DM).
This obviously has the disadvantage of complexity, but it can be fun nevertheless, and gives you and your girlfriend an idea of the different types of characters avaiable to play.
You could also combine 1 & 2 in the following way:
3) Start Phandelver with two (N)PCs, tone things down at the beginning, then add (N)PCs and enemies as you gain confidence
For example, your girlfriend could control one PC, and you one supporting NPC (just choose two from the pregenerated characters).
Spoiler, in case your girlfriend's reading :-)
Cut all encounters in half - so two goblins on the road, half the numbers of goblins in the cave, and either weaken the bugbear in Cragmaw Hideout and/or take his wolf away. Don't forget to halve the XP. Then at the end of Cragmaw Hideout, take Sildar Hallwinter with you and have him stay as a friendly (and tough) NPC. This presumes that your girlfriend will choose to go to Cragmaw Hideout first, but that is the most likely course of action, and you can always guide her back there from Phandalin if she goes there first.
Then as your girlfriend gains confidence playing, she could start to control another PC, and so you will have a tough enough party to face the later more challenging stuff without cutting things down.
For options 1-3 I strongly suggest you have a cleric in your party - parties with clerics last longer!
I haven't tried this myself, but for options 1-3 you might also want to read Play it Solo:Dungeons & Dragons Starter Set. Although it is geared to playing completely solo, looks like it might have some useful advice for converting Phandelver for playing with fewer players.
Or, for something completely different:
4) Find an adventure for one or two players from an earlier edition of D&D and convert it.
This is not my idea, but is described in more detail here:
Solo or 2 player adventures for 5e
You might have trouble doing this conversion just with the basic rules though - you would probably need at least the 5e Monster Manual, though I suppose you could just take the idea (and maps) from such an adventure and swap in monsters from the Starter Set.
Use your character's backgrounds to build intrinsic motivation for your players' characters.
If you're looking at the LMoP adventure, you hopefully have the pre-built characters handy. If you look at their backgrounds, you can see that for every character, there is some kind of intrinsic motivation built into their backgrounds for at least one of the elements of the adventure.
This provides the characters with clear motivation for why they should care about both the mission and the results of the mission.
You can do this in a custom setting in a couple of different ways. First, you could assign different background elements to your players, this may be a bit of a deal breaker for some players, but I've had good luck with it using a pre-built adventure and giving my players the option of which of 10 or so to select (Horde of the Dragon Queen provides 10 different optional background elements).
However, the best thing you can do is to pick up plot elements from your characters' backgrounds. Have them write a short paragraph or even just be super detailed when they are building their BIFT part of their background so that you can mine that for details and NPCs and other motivations.
If you're looking to provide intrinsic motivation, having it worked into your characters' backgrounds is the best and easiest way to do it.
Best Answer
I've been running the same starter set group with 7 players for the last 2 years and we're almost finished the last cave. It was also the first time many of us had ever played - I'd certainly never ran a game before.
Here goes. It's going to be a long one...
Basically think in terms of action economy.
If there's only a few goblins, add a few more (though be wary of this before level 3 due to players max hp). If you misjudged and it looks like it's going south, have some of them "retreat for reinforcements". Matt Colville's YouTube videos are your friend.
One problem you may find (or maybe it's just my rowdy rabble) is a large combat can take almost the entire session. My group gets one decent encounter done, one room of a dungeon (more if there're no enemies), or a fair amount of role play - but never more than any one or two combined in a session.
If there's already a large number of enemies consider upgrading to their stronger versions. Make one of the Goblins a Goblin Boss, a Bugbear into a Bugbear Chief, etc. This will in turn provide better exp and actually get them levelling up in decent time, as your total is now being divided by essentially double the recommended amount.
I have to say I "upgraded" more than I "swarmed". Players get less bored and they get to level up more often. I'm always looking for the highest CR I can ever throw at them.
Which highlights another problem you may run into.
Instead of having a group of 4 guys at level 5 reaching the last dungeon, it'll maybe be 7 guys at level 3. Which you think may equate to the same thing, but I'd worry about their max hp. If they're going down in one or two hits you may have a lethal game of dominoes on your hands if you don't stock them with extra potions beforehand.
Third thing I found useful...
Have them do some extra side quests not in the book. If there's a cool Hydra mini you have lying around, have something on the quest board to maybe clear a Hydra out of a nearby swamp or forest? Making my own content seemed impossible when I first began, but the adventure lays good foundation for inspiration (and most importantly confidence) to sprout and take form.
Or have a Hill Giant or two attack the town - nice bit of foreshadowing for running Storm King's Thunder afterwards? ;)
Not keen on creating your own content? Well...
When in Thundertree, one player said "What's in that volcano? DM said the eruption was magical - there must be something cool up there!"
Oh, drat! What will I do? There's nothing in the book about that. I know! I'll run White Plume Mountain (from TftYP)! Why do they want to go in? I'll have a Fire Giant in the crater that can't get past the 1st riddle - he'll offer to smith them some obsidian gear if they can clear the place out (with the option of a double-cross when they see him again)!
They just turned level 5 as they entered (I gave them the exp for successfully negotiating a way past the FG without fighting him) and they just turned level 6 as they left - before a Copper Dragon lands, a fight between the two ensues and the players have to decide who to join in and help...
And all before Cragmaw Castle. Now they're halfway thru the final dungeon, now modded and far tougher than the one originally published (Skeletons become Wights, Zombies are Ghouls/Ghasts, etc.), level 7, only two of the original party left (they like rolling up new characters), many deaths, and many emotional moments. It's been a wild 2 years!
You just have to mostly play it by ear and get a feel for it. If you're new to DMing like me it'll seem quite daunting at first, but I promise you'll soon have a good feel for things before too long.
Sorry for such a long answer. I just saw someone in pretty much the exact same boat as me when I started. Hope it helps. And good gaming!